Equilateral triangle side lengths

Question

There is a large right isosceles triangle with a hypotenuse length of 24. Inside the triangle is an equilateral triangle with a vertex on the midpoint of the hypotenuse. If the length of each side of the equilateral triangle is k((√3)-1), find k. 

 

I am so confused as to how to do this problem. 

Answer 1

Graphically interpreting the problem statement requirements as you have described, please view the figure at the following URL...

https://www.wyzant.com/resources/files/440474/equilateral_triangle_inscribed_in_a_right_triangle

 

Use the figure I have illustrated and try to understand the steps. In general, the solution approach was as follows....

 

1.  Use the decimal value for "k((√3)-1)" which equals "0.732k" for ease of calculation, equilateral triangle side length.

2.  Break down the figure into additional smaller right triangles.  I have introduced lettered vertex points to indentify additional triangles constructed and used in the solution.

3.  We can easily calculate the values of all interior angles and assign to all of the appropriate triangles shown in the figure.  Recall that the sum of all interior angles of a triangle equals 180degrees.

4.  Use trigonometric properties of right triangles and solve for additional triangle side lengths (along side AB).

5.  Use the the side lengths calculated for triangle DEF and equate the hypotenuse calculated value to "0.732k", solve for "k" (solution is k=12).

 

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