{Geometry} Find length of the equilateral triangle

In summary, the problem involves finding the length of the side of an equilateral triangle, FGH, which is also the midpoint of another equilateral triangle, CDE, inside a circle. Using the properties of equilateral triangles and the radius of the circle, the side FGH can be expressed as Asqrt(B)-C where A, B, and C are positive integers. The correct solution can be found using the law of cosines.
  • #1
youngstudent16
59
1

Homework Statement


http://i.imgur.com/lnk7e0D.png CDE is an equilateral triangle inside a circle, with side length 16. FGH is also an equilateral triangle and F is the mid point of DE. Find the length of the side FGH. Should be expressed as Asqrt(B)-C. Where A B and C are positive integers [/B]

Homework Equations


The radius of the circle and properties of the equilateral triangle

The Attempt at a Solution


ThetaBig = ThetaSmall = 60

SideBig * sin(60deg)+SideSmall * sin(60deg) = 2 * radius

SideSmall = (2 * radius - SideBig * sin(60deg)) / sin(60deg)

SideSmall = 2 * radius / sin(60deg) - SideBig

This gave me fraction answer of 64/3 - 16 not in the correct form
 
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  • #2
youngstudent16 said:
SideBig * sin(60deg)+SideSmall * sin(60deg) = 2 * radius

According to the picture this is not the case. The left side is actually a little bit less than 2 * radius, because the smaller triangle does not touch the circle at the bottom middle.
 
  • #3
DEvens said:
According to the picture this is not the case. The left side is actually a little bit less than 2 * radius, because the smaller triangle does not touch the circle at the bottom middle.

Picture is human error on my part with bad paint skills
 
  • #4
youngstudent16 said:
Picture is human error on my part with bad paint skills

I don't think I made myself clear. You have worked out the combined height of the two triangles, and this is slightly less than the diameter of the circle.
 
  • #5
Since we are dealing with equilateral triangles, it's not clear what "Side Big" and "Side Small" are.
 
  • #6
SteamKing said:
Since we are dealing with equilateral triangles, it's not clear what "Side Big" and "Side Small" are.

Look at the picture. There is a big triangle and a small triangle.
 
  • #7
SteamKing said:
Since we are dealing with equilateral triangles, it's not clear what "Side Big" and "Side Small" are.

Side big is meant to be the side of the bigger triangle which is known as 16. The small side is the one I'm trying to find expressed as Asqrt(B)-C where they are all integers.
 
  • #8
DEvens said:
I don't think I made myself clear. You have worked out the combined height of the two triangles, and this is slightly less than the diameter of the circle.
Ah yes I ended up using law of cosines and got the correct answer thank you for the hint.
 

Related to {Geometry} Find length of the equilateral triangle

1. How do you find the length of an equilateral triangle?

To find the length of an equilateral triangle, you can use the Pythagorean Theorem, which states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In an equilateral triangle, all three sides are equal, so you can use this formula to solve for the length of one side.

2. What is the formula for finding the length of an equilateral triangle?

The formula for finding the length of an equilateral triangle is s = √(3A)/2, where s is the length of one side and A is the area of the triangle. This formula is derived from the fact that an equilateral triangle can be divided into two equal right triangles, and then using the Pythagorean Theorem to solve for the length of one side.

3. Can you use trigonometry to find the length of an equilateral triangle?

Yes, you can use trigonometry to find the length of an equilateral triangle. For example, you can use the sine function to solve for the length of one side. The formula for this would be s = 2Rsin(60°), where s is the length of one side and R is the radius of the circumscribed circle of the equilateral triangle.

4. How does the perimeter of an equilateral triangle relate to its side length?

The perimeter of an equilateral triangle is equal to the sum of its three equal sides. So if you know the length of one side, you can simply multiply it by 3 to find the perimeter. Alternatively, you can use the formula P = 3s, where P is the perimeter and s is the length of one side.

5. Can the length of an equilateral triangle be found using only its angles?

No, the length of an equilateral triangle cannot be found using only its angles. In order to find the length, you need to know at least one side length or the area of the triangle.

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