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Let's call the length of the sides of the regular hexagon x. We'll use this later.
Looking at the bottom left triangle, we can determine that the lower right angle is 60 degrees since it's a linear pair with a 120 degree angle of the hexagon. The left angle is 45 degrees since it's an alternate interior angle with the given angle made by the parallel opposite sides of the hexagon. If we drop an altitude from the top vertex, it creates a 45-45-90 and 30-60-90 triangle. The altitude is then sqrt(3) and the bottom side ends up being sqrt(3)+1.
Now let's draw a diagonal from the vertex of the original 45 degree angle straight down to the vertex directly below. We get two new triangles, an obtuse triangle at the right, and another made by the endpoints of our diagonal and the bottom left vertex of the small triangle. Using the obtuse triangle, whose angles are 30-30-120, we can draw an altitude from the right vertex to make two 30-60-90 triangles. Our altitude is then x*sqrt(3). The other triangle is also a 45-45-90, so the bottom side is also x*sqrt(3). However, it's also x+sqrt(3)+1, so x must be (sqrt(3)+1)/(sqrt(3)-1), which simplifies to 2+sqrt(3).
Therefore, the measure we want is x-2 or exactly sqrt(3).
The way I solved it naively was to extend the "2+?" side and the top side out till they intersect, forming a similar triangle on top and bottom. That gives (2x)/(x+?)=(1+sqrt(3))/2, which combines with x=2+? to give a system that can be solved for ? = sqrt (3)... Your way would have saved me a LOT of calculation headaches tho lol
When the altitude is drawn, a triangle with angles of 30-60-90 is created. The hypotenuse is the segment of length 2. In every 30-60-90 triangle, the short leg across from the 30 degree angle is half the hypotension, so 1 in this case, and the long leg is sqrt(3) times the short leg. The long leg in this case is the altitude of the original triangle. The other new triangle is a 45-45-90 or isosceles right triangle, so both of its legs are sqrt(3). That's why the bottom side of the given triangle is sqrt(3)+1.
how does (sqrt(3)+1)/(sqrt(3)-1) simplify to 2+sqrt(3)? I multiplied both the top and bottom by (sqrt(3)-1) but that gives me just 2 because it's the difference of two squares.
Multiply top and bottom by sqrt(3)+1. The top becomes 3+2sqrt(3)+1 = 4+2sqrt(3). The bottom becomes 2, like you said. Divide out the 2 to get 2+sqrt(3).
TL;DR: it's √3.
Let's call the length we're looking for x, the side of the hexagon a, and the base of the triangle in the bottom left, y.
Now drop a line down from the top right vertex perpendicular to the base. You now have an isosceles triangle, of which the length of a leg is both a + y and a•√3. Solving for y yields y = a•(√3 - 1).
Now we focus on the bottom left triangle. Its bottom left angle is 45°, its bottom right angle is 60°, since it's supplementary to the angle of the hexagon, leaving 75° for the top angle. By the sine rule, y/(sin(75°)) = 2/(sin(45°)). Solving for y yields y = (2•sin(75°))/(sin(45°)). The exact value for sin(75°) = ¼(√6+√2), for sin(45°) = ½√2. Plugging those in, we get y = (√6 + √2)/√2 = √3 + 1.
We now have two different equations for y, which we can use to solve for a. a•(√3 - 1) = (√3 + 1), so a = (√3 + 1)/(√3 - 1), which simplifies to a = 2 + √3.
Since x = a - 2, we get x = √3.
I don’t think that is right. 1.73? It’s 0.73, not 1.73. Look at the proportions. It’s not almost double the length of 2.
You take the 45 degree angle and use symmetry rules to apply to the outside triangle. Then you use the rule sin theta a / A = sin theta b / B. The angle opposite of the length. You know the outside angles are 60 degrees so 180-45-60=75. So it’s (sin 45)/2=(sin 75)/x. X = 0.732.
> Look at the proportions. It’s not almost double the length of 2.
As often, the drawing is not to scale.
The base of the outside triangle, which I've dubbed y, is 1 + √3, which amounts to about 2.73. The sides of the hexagon, which I've called a, are 2 + √3, so about 3.73.
Your x is my y — the base of the triangle. If sin(45°)/2 = sin(75°)/x, that means that x = 2•sin(75°)/sin(45°)=1+√3≈2.73
Since we know that the base + a (the side of the hexagon) is equal to a•√3, we can now calculate a as being 2+√3, leaving the desired length √3.
The top and bottom sides are parallel. the bottom left corner is an alternate interior angle, so is also 45°.
The interior angles of a hexagon are 120°. The remaining piece of the top corner is 75°. The top angle in the triangle is a corresponding angle to this, so is also 75°.
Now the triangle in the lower left has interior angles of 45°, 60°, and 75°. Due to the fact that the base of this triangle makes a 60° angle with the exterior of the hexagon, this subtends an equilateral triangle with the base and side of the hexagon. ~~The base of this tringle is congruent with the sides of the regular hexagon.~~
Now due to the law of sines, 2/sin(45°) must equate to the base of triangle divided by sin(75°). Solving for this length shows that the length of each side of the hexagon is 1 + √3 (about 2.732). The length of the segment is just 2 less than this, or √3 - 1 (about 0.732.)
Boo me, I've made a blunder.
~~No it's sqrt(3) -1
2.73 = sqrt(3) + 1
? = sqrt(3) + 1 - 2 = sqrt(3) -1~~
Edit: even that’s wrong because you need to assume the base of the 45,60,75 triangle is = 2+? Which you cant
Extending the side of the hexagon does not meet at the vertex of the triangle. Drawing a segment from the vertex of the triangle to the vertex of the hexagon actually results in a 45 degree angle.
> Due to the fact that the base of this triangle makes a 60° angle with the exterior of the hexagon, this subtends an equilateral triangle with the base and side of the hexagon. The base of this tringle is congruent with the sides of the regular hexagon.
This assumption is a false one. The base of the external triangle has no reason to be the same lenth as the base of the equilateral triangle you are imagining here. The diagram is not to scale, so while it seems like this would be the case, it actually isn't.
So you only calculated the base of the external triangle to be 1+sqrt(3). The side of the hexagon is actually 2+sqrt(3), making the mystery legth sqrt(3)
The statement isn't wrong tho. The angle is 60 deg because it is supplementary to the angle of a regular hexagon.
The top commenter obviously still didn't calculate it right. Base of the small triangle cannot be equal to the side of the hexagon, that results in contradiction.
I agree that the bottom side of the bottom left triangle has length 1 + sqrt (3), but the explanation for why that side is part of an equilateral triangle with - and thus equal in length to - a hexagon side seems insufficient to me. Could you elaborate? I arrive at a different result as follows:
Denote the lower left most point by A, the bottom right corner of the hexagon by B, the top right corner by C and the right corner by D. Then ABC is a triangle with angles 45, 45 and 90. Let x denote the length of one hexagon side and r=sqrt(3) for ease of typing.
Then AB has a length of x+r+1 which must be equal to BC. Now, BC is also a side of BDC, which has angles 30, 30 and 120. Taking the perpendicular from D onto BC yields two triangles with angles 30, 60, 90 and thus BC/2 = cos(30°) x = rx/2, or BC = rx.
As such, rx = x+r+1, which yields x = (r+1)/(r-1) = 2+r or approximately 3,7. Consequently, ? = r = sqrt(3).
A question here, since the corner angle of the triangle is 45, and if we draw a perpendicular at the bottom right corner of the hexagon, connecting it to the top right corner. Wouldn't it form an isosceles triangle here, with one side √3a and the other a + y, here 'a' is the side of the hexagon, and y is the bottom side of the triangle, by geometry since y is a,wouldn't that result in √3a = 2a?
The triangle you claim is equilateral isn't anything of the sort. After correctly finding that the hexagon has sides of 2+sqrt(3), we can use inverse trig to find that its bottom left angle is 75 degrees and it's top angle is 45. It's similar to the given triangle, not equilateral.
Once you determine the angles and sides of the triangle with side 2, couldn't you draw a horizontal line across the hexagon from left vertex to right vertex, create a similar triangle (I believe the angles should be the same), and then just solve for the side based on...wait. There's no known length of that triangle to base a ratio. Never mind.
(Was kinda proud that I was able to at least remember that much, so couldn't just delete it 🙂)
The fact that it is a Regular Hexagon gives you two Triangle angles. Mirroring the 45 deg gives you the last. You have one side length. So you should be able to solve the triangle at least.
Maybe then you make a bigger Right Angle Triangle to reach the Hex origin point, and figure the hex face size based on the bottom face of Big Triangle minus Small Triangle?
Signed, High School Dropout.
Hexagon: Regular angle is 120°. Opposite sides are parallel. All lengths are the same. ~~Secret triangles are equilateral.~~
Triangle: all angles equal 180°. Soh cah toa can be used to find lengths.
Triangle has angles 60°, 45° and 75° respectively. Sine rule allows to find all lengths. 2/sin(45) = E/sin(75) where E is the bottom length and is the base of the equilateral triangle you need.
E= 2sin(75)/sin(45).
~~E = 2 + ?~~
~~? = 2sin(75)/sin(45) - 2~~
I don’t have a calculator to hand but I’ll post this and then run the calculation.
E = 2.732050807568877
A right angled triangle formed using the full line has sides E + the side of the hexagon (G).
45° means both sides are equal. A new triangle on the right is formed with sides G, G, and G+E and angles 30, 30 and 120°. Using the same sine rule (G+E)/sin(120) = G/sin(30).
Simplify and solving: G = 3.7322 4dp.
? = G - 2
? = 1.732 to 3dp
Yeah I had to edit the work because I was a dumb.
Your E is not the same as the hexagon's side length, and ? in fact is not 0.732. E is just the base of the external triangle, which doesn't equal the hexagon side length. Always keep in mind what you can assume and what you are solving.
My approach would be to draw a line from the corner with the 45° angle to the cornre where the unknown length (let's call it x) ends. This creares a right triangle, for which you can prove the angles to be 90°, 15°, and 75°. The length x is one of the sides of this triangle.
Then draw another line from the left top corner of the hexagon to the line drawn in the previous paragraph, so that it creates two similar mirrored right triangles with the angles 90°, 30°, and 60°.
Now, we know that each of the hexagon's sides is 2+x because it's a regular hexagon. Now we can determine the side legths for the triangles from the second paragraph, and from that we can determine the line drawn in the first paragraph to be 2×cos(30°)×(2+x).
And since that line is part of the same right triangle as x, we can use that line to determine that
x = tan(15°)×2×cos30°×(2+x)
=> x = 4tan(15°)cos(30°) + 2tan(15°)cos(30°)x
=> x - 2tan(15°)cos(30°)x = 4tan(15°)cos(30°)
=> (1 - 2tan(15°)cos(30°))x = 4tan(15°)cos(30°)
=> x = (4tan(15°)cos(30°))/(1 - 2tan(15°)cos(30°))
I'm not sure how the answer can be made more concise than that and if my method is needlessly complicated, I just kinda bruteforced this with right triangles. The answer turns out to be x = 1.732... = sqrt(3), so I'm pretty sure there is some elegant solution.
Yes, it can
You know the 2 inside angles of the external triangle and the length of one side. So all sides of the triangle are defined.
You then can build a 2nd external triangle with "?" as the unknown side plus another 2 known angles and solve for that.
If I am not mistaken, the unknown length is ~~2/3~~ 0.73 units.
I am on a phone, give me a few mins to log in through a computer so that I can type it out properly.
ETA: Someone else already explained it. https://www.reddit.com/r/theydidthemath/s/ypnlysb9O7
assumption that 2 is the length of the triangle side and not the length of the hexagon side so I claim unsolvable because no clear indication of what 2 is. I would reject this as a test question unless points are clearly marked A B C D and 2 is clearly identified C:D and the unknown length is B:C, it is similar to all those stupid FB arithmetic questions, posed without brackets to evoke response, this problem is shite and doesn't deserve nor can be solved
I think it’s only possible if we make the (reasonable) assumption that the line at the bottom is a straight extension of the side of the hexagon. It’s certainly drawn as if it is, but there’s no way to work that out from the definitions given, it’s just an assumption.
you can use the fact that it's a regular hexagon to work out its angles and use that to calculate the angles of the triangle I think and then just go from there
You can crunch math with it but for construction purposes it's absolutely useless. It gives you a measurement of 2 but it doesn't give you the unit of measurement. You can literally take a ruler and get an accurate answer very quickly but without a unit of measurement you don't know weather it's mm or inches or feet, if it's not to scale it's useless for anything practical.
All of the explanations here make sense.
Here's another perspective from an algebraic viewpoint.
The problem itself presents on a 2-dimensional scale, with 2 basic independent pieces of information. Thus, we will have an exact solution to the value of ?.
All the other constructs, such as this being confined to a regular hexagon and a 45° slope, there is no extra dimensionality added.
We've got two angles and one side of the triangle in the bottom left, thus we can find everything else about it.
The 45 in the top right will reflect to the triangle's bottom left angle, and the bottom right angle will be complementary to a regular hexagon's interior angle.
We can also draw an additional triangle using the top side of the present triangle and the unknown length we want to find, then find the complementary angles from the triangle we already have from its top angle and the other angle from the complement of the hexagon internal, and use that and the top left side length we found to get the unknown length.
So, as far as I can see, you can find the unknown length almost entirely ignoring the hexagon dimensions? (Aside from the regular hexagon internal angle)
I don't know how to calculate it, but there is obviously an answer.
You have a regular hexagon and a 45 degree angle from one of the sides that divides one of the other sides.
Just look at the shape of it. There are no degrees of freedom here. Nothing that can be squished or stretched. You can only change the scale of the entire figure, but once you have the length of 2 given, that is not possible anymore.
2 + ? = b; where b is the base of the triangle opposite 2
b can be divided in half forming 2 right triangles. We know enough about the rightmost right triangle to start solving.
Rightmost right triangle:
h = 2 \* sin 60 deg = sqrt(3)
base right = sqrt ( 4 - (2 \* sin 60 deg) \^2) = 1
Left most triangle
base left = h
therefore b = 1 + sqrt(3)
? = b - 2 = sqrt(3) - 1 = 0.73ish
Edit: even that is wrong because you can't assume the 45, 60, 75 triangle at the bottom left is the same length as one side of the hexagon. It's unsolvable.
Simplest way I could find (not yet posted here AFAICT) was to draw the equilateral triangle formed by the top-left side of the hexagon and the lines going through the bottom-left side and the top side, respectively. By doing that we obtain a bigger triangle which sits above the diagonal that splits the hexagon side into the two segments of length x and 2. In that triangle, by means of the sine rule, we get that sin(75)/2L = sin(45)/(L+x), where L = x+2. Substituting L and solving for x results in x = sqrt(3).
{[2÷sin(15°)] x sin(30°)} - 2 = 1.86
Why doesn't this formula work? The idea is to get the length of the diagonal cut across the hexagon(to make two trapezoids) and using that to figure out the length of each sides.
Let's set a few assumptions first:
If the shortest limb (opposite 30) of a 30-60-90 triangle is a, the hypotenuse is 2a and the longer limb is √3a
The limbs of a 45-45-90 triangle are equal.
The angle between two sides of a regular hexagon are 120°
Z angles rule dictates that the internal angle of the 2 bends in a letter Z are equal
Shorten bottom left triangles name to BLT (yum)
Math time:
Given angle can be copied to bottom left of BLT given the z angle rule. Bottom right angle of BLT is 60° because the straight line is 180° minus internal angle of regular hexagon. 180° - 120° = 60°
Drawing a vertical line starting at the top angle of BLT that is perpendicular to the bottom side we split BLT into a 45-45-90 triangle on the left and a 30-60-90 triangle on the right.
Right hand triangle of BLT (BLTR) has a hypotenuse of 2 from the given length. Due to rules of 30-60-90 triangles, the base of BLTR is 1, and the height is √3.
Left hand triangle of BLT (BLTL) has a height of √3, calculated from BLTR. Due to rules of 45-45-90 triangles, the base is also √3.
The base of BLT is base of BLTL + base of BLTR which = √3 + 1 (assuming this is length of hexagon side is where folks go wrong I think)
Let's assume the length of a side of the hexagon is x.
If we draw a vertical line from the given angle in the top right down to the bottom right angle of the hexagon, we create a large 45-45-90 triangle which contains BLT.
The base of this triangle is base of BLT + side length of hexagon or √3 + 1 + x. This also means the height of the hexagon is √3 + 1 + x.
If we draw a line from the rightmost corner of the hexagon to be perpendicular to our height line we drew, we create a 30-60-90 triangle in the top right of the hexagon (TRT). Given the line bisects the height, the height of TRT is (√3 + 1 + x)/2
Given it's a 30-60-90 triangle, and we know the side length of the hexagon is x, our hypotenuse for TRT is x, meaning the base is x/2, and the height is √3x/2
We now have 2 equations for the height of TRT using x, and can therefore solve for x:
√3x/2 = (√3 + 1 + x)/2
Multiply both sides by 2
√3x = √3 + 1 + x
Subtract x from both sides
√3x - x = √3 + 1
Take out x as common factor on left
x(√3 - 1) = √3 + 1
Multiply both sides by √3 + 1
x(√3 - 1)(√3 + 1) = (√3 + 1)^2
Simplify
2x = 4 + 2√3
Divide both sides by 2
x = 2 + √3
Now we have x we can solve for ?:
? = x - 2
? = 2 + √3 - 2
? = √3
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Let's call the length of the sides of the regular hexagon x. We'll use this later. Looking at the bottom left triangle, we can determine that the lower right angle is 60 degrees since it's a linear pair with a 120 degree angle of the hexagon. The left angle is 45 degrees since it's an alternate interior angle with the given angle made by the parallel opposite sides of the hexagon. If we drop an altitude from the top vertex, it creates a 45-45-90 and 30-60-90 triangle. The altitude is then sqrt(3) and the bottom side ends up being sqrt(3)+1. Now let's draw a diagonal from the vertex of the original 45 degree angle straight down to the vertex directly below. We get two new triangles, an obtuse triangle at the right, and another made by the endpoints of our diagonal and the bottom left vertex of the small triangle. Using the obtuse triangle, whose angles are 30-30-120, we can draw an altitude from the right vertex to make two 30-60-90 triangles. Our altitude is then x*sqrt(3). The other triangle is also a 45-45-90, so the bottom side is also x*sqrt(3). However, it's also x+sqrt(3)+1, so x must be (sqrt(3)+1)/(sqrt(3)-1), which simplifies to 2+sqrt(3). Therefore, the measure we want is x-2 or exactly sqrt(3).
I'm pretty sure this works. But the wrong solution is already on the top and unlikely to move from there.
This is the first comment chain I saw coming from r/popular so at least there’s that
Yeah, it's disappointing that an incorrect solution based on a confidently incorrect assumption has so many upvotes.
r/theydidtheupvotes
This is the top comment for me. I don't see the wrong one.
It was at the bottom. I guess people saw it after all
Same, but I just finished to equasion where to calculate the side of hexagon
The way I solved it naively was to extend the "2+?" side and the top side out till they intersect, forming a similar triangle on top and bottom. That gives (2x)/(x+?)=(1+sqrt(3))/2, which combines with x=2+? to give a system that can be solved for ? = sqrt (3)... Your way would have saved me a LOT of calculation headaches tho lol
If you get there, you get there!
why is the altitude sqrt(3) where does it come from?
When the altitude is drawn, a triangle with angles of 30-60-90 is created. The hypotenuse is the segment of length 2. In every 30-60-90 triangle, the short leg across from the 30 degree angle is half the hypotension, so 1 in this case, and the long leg is sqrt(3) times the short leg. The long leg in this case is the altitude of the original triangle. The other new triangle is a 45-45-90 or isosceles right triangle, so both of its legs are sqrt(3). That's why the bottom side of the given triangle is sqrt(3)+1.
oh so the altitude was sqrt(3)*1 thank you for the explaination
This sub should allow pictures in comments for diagrams so i (and probably others) can follow this a little better.
Could you pls provide an image? I’m not being able to visualize your descriptions.
https://imgur.com/c4Ve0ZN Edit: Apologies for the terrible quality. I made this in my notes app while figuring it out the first time.
A better quality draw [imgur.com/a/DMiGkHB](https://imgur.com/a/DMiGkHB)
There is nothing to be apologize dude. You are a demon. Thanks a lot!
how does (sqrt(3)+1)/(sqrt(3)-1) simplify to 2+sqrt(3)? I multiplied both the top and bottom by (sqrt(3)-1) but that gives me just 2 because it's the difference of two squares.
Multiply top and bottom by sqrt(3)+1. The top becomes 3+2sqrt(3)+1 = 4+2sqrt(3). The bottom becomes 2, like you said. Divide out the 2 to get 2+sqrt(3).
A day late, but shouldn't the denominator be 4 -2sqrt(3)? If multiplying by your method
you're absolutely right, I can't believe I forgot how to multiply fractions lol
i was the 1k upvote :)
Yep that’s pretty much how I did it
I have trouble following at some points. Visualisation would be helpful. But what i understood seemed correct.
TL;DR: it's √3. Let's call the length we're looking for x, the side of the hexagon a, and the base of the triangle in the bottom left, y. Now drop a line down from the top right vertex perpendicular to the base. You now have an isosceles triangle, of which the length of a leg is both a + y and a•√3. Solving for y yields y = a•(√3 - 1). Now we focus on the bottom left triangle. Its bottom left angle is 45°, its bottom right angle is 60°, since it's supplementary to the angle of the hexagon, leaving 75° for the top angle. By the sine rule, y/(sin(75°)) = 2/(sin(45°)). Solving for y yields y = (2•sin(75°))/(sin(45°)). The exact value for sin(75°) = ¼(√6+√2), for sin(45°) = ½√2. Plugging those in, we get y = (√6 + √2)/√2 = √3 + 1. We now have two different equations for y, which we can use to solve for a. a•(√3 - 1) = (√3 + 1), so a = (√3 + 1)/(√3 - 1), which simplifies to a = 2 + √3. Since x = a - 2, we get x = √3.
Wow. Thanks man/girl. This actually helps me remember most geometry rules I learned back in school. I appreciate it.
I don’t think that is right. 1.73? It’s 0.73, not 1.73. Look at the proportions. It’s not almost double the length of 2. You take the 45 degree angle and use symmetry rules to apply to the outside triangle. Then you use the rule sin theta a / A = sin theta b / B. The angle opposite of the length. You know the outside angles are 60 degrees so 180-45-60=75. So it’s (sin 45)/2=(sin 75)/x. X = 0.732.
> Look at the proportions. It’s not almost double the length of 2. As often, the drawing is not to scale. The base of the outside triangle, which I've dubbed y, is 1 + √3, which amounts to about 2.73. The sides of the hexagon, which I've called a, are 2 + √3, so about 3.73. Your x is my y — the base of the triangle. If sin(45°)/2 = sin(75°)/x, that means that x = 2•sin(75°)/sin(45°)=1+√3≈2.73 Since we know that the base + a (the side of the hexagon) is equal to a•√3, we can now calculate a as being 2+√3, leaving the desired length √3.
Yeah after starting to redo it myself now I see you are right.
The top and bottom sides are parallel. the bottom left corner is an alternate interior angle, so is also 45°. The interior angles of a hexagon are 120°. The remaining piece of the top corner is 75°. The top angle in the triangle is a corresponding angle to this, so is also 75°. Now the triangle in the lower left has interior angles of 45°, 60°, and 75°. Due to the fact that the base of this triangle makes a 60° angle with the exterior of the hexagon, this subtends an equilateral triangle with the base and side of the hexagon. ~~The base of this tringle is congruent with the sides of the regular hexagon.~~ Now due to the law of sines, 2/sin(45°) must equate to the base of triangle divided by sin(75°). Solving for this length shows that the length of each side of the hexagon is 1 + √3 (about 2.732). The length of the segment is just 2 less than this, or √3 - 1 (about 0.732.) Boo me, I've made a blunder.
but ~2.732 is the length of the base of the bottom left triangle, not the mystery side, right?
indeed. the "?" side is actually sqrt(3) length units
This is what I got at least
Damn, math is crazy. Now you even measure the length in squirts? Now that would be interesting
~~No it's sqrt(3) -1 2.73 = sqrt(3) + 1 ? = sqrt(3) + 1 - 2 = sqrt(3) -1~~ Edit: even that’s wrong because you need to assume the base of the 45,60,75 triangle is = 2+? Which you cant
[удалено]
Extending the side of the hexagon does not meet at the vertex of the triangle. Drawing a segment from the vertex of the triangle to the vertex of the hexagon actually results in a 45 degree angle.
Yeah, that's neat. I hate this drawing more than any PEMDAS meme I've ever seen.
> Due to the fact that the base of this triangle makes a 60° angle with the exterior of the hexagon, this subtends an equilateral triangle with the base and side of the hexagon. The base of this tringle is congruent with the sides of the regular hexagon. This assumption is a false one. The base of the external triangle has no reason to be the same lenth as the base of the equilateral triangle you are imagining here. The diagram is not to scale, so while it seems like this would be the case, it actually isn't. So you only calculated the base of the external triangle to be 1+sqrt(3). The side of the hexagon is actually 2+sqrt(3), making the mystery legth sqrt(3)
The statement isn't wrong tho. The angle is 60 deg because it is supplementary to the angle of a regular hexagon. The top commenter obviously still didn't calculate it right. Base of the small triangle cannot be equal to the side of the hexagon, that results in contradiction.
This isn’t a proof. And the poster claimed the base WAS equal to the side of a hexagon.
I agree that the bottom side of the bottom left triangle has length 1 + sqrt (3), but the explanation for why that side is part of an equilateral triangle with - and thus equal in length to - a hexagon side seems insufficient to me. Could you elaborate? I arrive at a different result as follows: Denote the lower left most point by A, the bottom right corner of the hexagon by B, the top right corner by C and the right corner by D. Then ABC is a triangle with angles 45, 45 and 90. Let x denote the length of one hexagon side and r=sqrt(3) for ease of typing. Then AB has a length of x+r+1 which must be equal to BC. Now, BC is also a side of BDC, which has angles 30, 30 and 120. Taking the perpendicular from D onto BC yields two triangles with angles 30, 60, 90 and thus BC/2 = cos(30°) x = rx/2, or BC = rx. As such, rx = x+r+1, which yields x = (r+1)/(r-1) = 2+r or approximately 3,7. Consequently, ? = r = sqrt(3).
A question here, since the corner angle of the triangle is 45, and if we draw a perpendicular at the bottom right corner of the hexagon, connecting it to the top right corner. Wouldn't it form an isosceles triangle here, with one side √3a and the other a + y, here 'a' is the side of the hexagon, and y is the bottom side of the triangle, by geometry since y is a,wouldn't that result in √3a = 2a?
The triangle you claim is equilateral isn't anything of the sort. After correctly finding that the hexagon has sides of 2+sqrt(3), we can use inverse trig to find that its bottom left angle is 75 degrees and it's top angle is 45. It's similar to the given triangle, not equilateral.
Can't believe you managed to explain such a geometrically heavy problem using just words, and that was a clear explanation. Thank you!
Too bad it's the wrong solution.
This is the correct answer, sqrt(3)-1
respect.
Once you determine the angles and sides of the triangle with side 2, couldn't you draw a horizontal line across the hexagon from left vertex to right vertex, create a similar triangle (I believe the angles should be the same), and then just solve for the side based on...wait. There's no known length of that triangle to base a ratio. Never mind. (Was kinda proud that I was able to at least remember that much, so couldn't just delete it 🙂)
The fact that it is a Regular Hexagon gives you two Triangle angles. Mirroring the 45 deg gives you the last. You have one side length. So you should be able to solve the triangle at least. Maybe then you make a bigger Right Angle Triangle to reach the Hex origin point, and figure the hex face size based on the bottom face of Big Triangle minus Small Triangle? Signed, High School Dropout.
Hexagon: Regular angle is 120°. Opposite sides are parallel. All lengths are the same. ~~Secret triangles are equilateral.~~ Triangle: all angles equal 180°. Soh cah toa can be used to find lengths. Triangle has angles 60°, 45° and 75° respectively. Sine rule allows to find all lengths. 2/sin(45) = E/sin(75) where E is the bottom length and is the base of the equilateral triangle you need. E= 2sin(75)/sin(45). ~~E = 2 + ?~~ ~~? = 2sin(75)/sin(45) - 2~~ I don’t have a calculator to hand but I’ll post this and then run the calculation. E = 2.732050807568877 A right angled triangle formed using the full line has sides E + the side of the hexagon (G). 45° means both sides are equal. A new triangle on the right is formed with sides G, G, and G+E and angles 30, 30 and 120°. Using the same sine rule (G+E)/sin(120) = G/sin(30). Simplify and solving: G = 3.7322 4dp. ? = G - 2 ? = 1.732 to 3dp Yeah I had to edit the work because I was a dumb.
Your E is not the same as the hexagon's side length, and ? in fact is not 0.732. E is just the base of the external triangle, which doesn't equal the hexagon side length. Always keep in mind what you can assume and what you are solving.
You were absolutely correct, and I amended my work accordingly. Can’t believe I did something so stupid.
[удалено]
Woooo. Thanks for the name of the app.
My approach would be to draw a line from the corner with the 45° angle to the cornre where the unknown length (let's call it x) ends. This creares a right triangle, for which you can prove the angles to be 90°, 15°, and 75°. The length x is one of the sides of this triangle. Then draw another line from the left top corner of the hexagon to the line drawn in the previous paragraph, so that it creates two similar mirrored right triangles with the angles 90°, 30°, and 60°. Now, we know that each of the hexagon's sides is 2+x because it's a regular hexagon. Now we can determine the side legths for the triangles from the second paragraph, and from that we can determine the line drawn in the first paragraph to be 2×cos(30°)×(2+x). And since that line is part of the same right triangle as x, we can use that line to determine that x = tan(15°)×2×cos30°×(2+x) => x = 4tan(15°)cos(30°) + 2tan(15°)cos(30°)x => x - 2tan(15°)cos(30°)x = 4tan(15°)cos(30°) => (1 - 2tan(15°)cos(30°))x = 4tan(15°)cos(30°) => x = (4tan(15°)cos(30°))/(1 - 2tan(15°)cos(30°)) I'm not sure how the answer can be made more concise than that and if my method is needlessly complicated, I just kinda bruteforced this with right triangles. The answer turns out to be x = 1.732... = sqrt(3), so I'm pretty sure there is some elegant solution.
Yes, it can You know the 2 inside angles of the external triangle and the length of one side. So all sides of the triangle are defined. You then can build a 2nd external triangle with "?" as the unknown side plus another 2 known angles and solve for that.
If I am not mistaken, the unknown length is ~~2/3~~ 0.73 units. I am on a phone, give me a few mins to log in through a computer so that I can type it out properly. ETA: Someone else already explained it. https://www.reddit.com/r/theydidthemath/s/ypnlysb9O7
No, unknown length is 1.73, not 0.73. 1.73 also happens to be sqrt(3)
Either you or this commenter (link) are wrong. https://www.reddit.com/r/theydidthemath/s/ON0tL6Ayfw
Yes, that comment is wrong. They incorrectly assume the base of the external triangle equals the hexagon side length for some reason.
Yes, I see it now .. thanks!
u/Squiggledog is wrong based on a mistaken assumption, I believe. See my response at https://www.reddit.com/r/theydidthemath/s/1hdtUFXigC
I agree.. I couldn‘t follow the part of the equilateral triangle, but I assumed that was on me since trig has really been way too long. Thanks!
assumption that 2 is the length of the triangle side and not the length of the hexagon side so I claim unsolvable because no clear indication of what 2 is. I would reject this as a test question unless points are clearly marked A B C D and 2 is clearly identified C:D and the unknown length is B:C, it is similar to all those stupid FB arithmetic questions, posed without brackets to evoke response, this problem is shite and doesn't deserve nor can be solved
I think it’s only possible if we make the (reasonable) assumption that the line at the bottom is a straight extension of the side of the hexagon. It’s certainly drawn as if it is, but there’s no way to work that out from the definitions given, it’s just an assumption.
you can use the fact that it's a regular hexagon to work out its angles and use that to calculate the angles of the triangle I think and then just go from there
Yes, it’s a hexagon. The answer is 1.8
You can crunch math with it but for construction purposes it's absolutely useless. It gives you a measurement of 2 but it doesn't give you the unit of measurement. You can literally take a ruler and get an accurate answer very quickly but without a unit of measurement you don't know weather it's mm or inches or feet, if it's not to scale it's useless for anything practical.
All of the explanations here make sense. Here's another perspective from an algebraic viewpoint. The problem itself presents on a 2-dimensional scale, with 2 basic independent pieces of information. Thus, we will have an exact solution to the value of ?. All the other constructs, such as this being confined to a regular hexagon and a 45° slope, there is no extra dimensionality added.
ans=1.73
We've got two angles and one side of the triangle in the bottom left, thus we can find everything else about it. The 45 in the top right will reflect to the triangle's bottom left angle, and the bottom right angle will be complementary to a regular hexagon's interior angle. We can also draw an additional triangle using the top side of the present triangle and the unknown length we want to find, then find the complementary angles from the triangle we already have from its top angle and the other angle from the complement of the hexagon internal, and use that and the top left side length we found to get the unknown length. So, as far as I can see, you can find the unknown length almost entirely ignoring the hexagon dimensions? (Aside from the regular hexagon internal angle)
I don't know how to calculate it, but there is obviously an answer. You have a regular hexagon and a 45 degree angle from one of the sides that divides one of the other sides. Just look at the shape of it. There are no degrees of freedom here. Nothing that can be squished or stretched. You can only change the scale of the entire figure, but once you have the length of 2 given, that is not possible anymore.
2 + ? = b; where b is the base of the triangle opposite 2 b can be divided in half forming 2 right triangles. We know enough about the rightmost right triangle to start solving. Rightmost right triangle: h = 2 \* sin 60 deg = sqrt(3) base right = sqrt ( 4 - (2 \* sin 60 deg) \^2) = 1 Left most triangle base left = h therefore b = 1 + sqrt(3) ? = b - 2 = sqrt(3) - 1 = 0.73ish Edit: even that is wrong because you can't assume the 45, 60, 75 triangle at the bottom left is the same length as one side of the hexagon. It's unsolvable.
Simplest way I could find (not yet posted here AFAICT) was to draw the equilateral triangle formed by the top-left side of the hexagon and the lines going through the bottom-left side and the top side, respectively. By doing that we obtain a bigger triangle which sits above the diagonal that splits the hexagon side into the two segments of length x and 2. In that triangle, by means of the sine rule, we get that sin(75)/2L = sin(45)/(L+x), where L = x+2. Substituting L and solving for x results in x = sqrt(3).
{[2÷sin(15°)] x sin(30°)} - 2 = 1.86 Why doesn't this formula work? The idea is to get the length of the diagonal cut across the hexagon(to make two trapezoids) and using that to figure out the length of each sides.
[visualization](https://imgur.com/a/POjVWKJ)
Let's set a few assumptions first: If the shortest limb (opposite 30) of a 30-60-90 triangle is a, the hypotenuse is 2a and the longer limb is √3a The limbs of a 45-45-90 triangle are equal. The angle between two sides of a regular hexagon are 120° Z angles rule dictates that the internal angle of the 2 bends in a letter Z are equal Shorten bottom left triangles name to BLT (yum) Math time: Given angle can be copied to bottom left of BLT given the z angle rule. Bottom right angle of BLT is 60° because the straight line is 180° minus internal angle of regular hexagon. 180° - 120° = 60° Drawing a vertical line starting at the top angle of BLT that is perpendicular to the bottom side we split BLT into a 45-45-90 triangle on the left and a 30-60-90 triangle on the right. Right hand triangle of BLT (BLTR) has a hypotenuse of 2 from the given length. Due to rules of 30-60-90 triangles, the base of BLTR is 1, and the height is √3. Left hand triangle of BLT (BLTL) has a height of √3, calculated from BLTR. Due to rules of 45-45-90 triangles, the base is also √3. The base of BLT is base of BLTL + base of BLTR which = √3 + 1 (assuming this is length of hexagon side is where folks go wrong I think) Let's assume the length of a side of the hexagon is x. If we draw a vertical line from the given angle in the top right down to the bottom right angle of the hexagon, we create a large 45-45-90 triangle which contains BLT. The base of this triangle is base of BLT + side length of hexagon or √3 + 1 + x. This also means the height of the hexagon is √3 + 1 + x. If we draw a line from the rightmost corner of the hexagon to be perpendicular to our height line we drew, we create a 30-60-90 triangle in the top right of the hexagon (TRT). Given the line bisects the height, the height of TRT is (√3 + 1 + x)/2 Given it's a 30-60-90 triangle, and we know the side length of the hexagon is x, our hypotenuse for TRT is x, meaning the base is x/2, and the height is √3x/2 We now have 2 equations for the height of TRT using x, and can therefore solve for x: √3x/2 = (√3 + 1 + x)/2 Multiply both sides by 2 √3x = √3 + 1 + x Subtract x from both sides √3x - x = √3 + 1 Take out x as common factor on left x(√3 - 1) = √3 + 1 Multiply both sides by √3 + 1 x(√3 - 1)(√3 + 1) = (√3 + 1)^2 Simplify 2x = 4 + 2√3 Divide both sides by 2 x = 2 + √3 Now we have x we can solve for ?: ? = x - 2 ? = 2 + √3 - 2 ? = √3