Thread: Equilateral triangles

hi guys

I need some help in solving this problem and another problem both extremely similar and both involving the use of the fraction sqrt3/2. I do not undeerstand this concept.

btw here is my question

In An equilateral triangle ABC of side x, a inscibed circle is drawn and then a cyclic square is drawn circumscribed by the circle. Find the area of the square

my other question is similar

in an equilateral triangle ABC of sides 2x, a rectangle is constructed with the midpoints AB and AC as two vertices and the other two lying on bc. I understand one side is x but how do you find the breadth

thnx in advance

Originally Posted by bloo.tomarto

hi guys

I need some help in solving this problem and another problem both extremely similar and both involving the use of the fraction sqrt3/2. I do not undeerstand this concept.

btw here is my question

In An equilateral triangle ABC of side x, a inscibed circle is drawn and then a cyclic square is drawn circumscribed by the circle. Find the area of the square

Dropping a perpendicular from the vertex of an equilateral triangle, with side length x, to the opposite side gives you two right triangles, each having hypotenuse of length x and one leg of length x/2. By the Pythagorean theorem, the other leg, the altitude of the equilateral triangle, is . Taking the distance from the vertex to the center of the circle, a radius of the circle and a diagonal of the square, to be y, the distance from the center of the circle to the opposite side is and now we have a right triangle with one leg of length , one leg of length x/2, and hypotenuse of length y. Set up the Pythagorean theorem for that triangle, solve for y. Now use the fact that is a square has diagonal of length y, then its side has length to find the area of the square.

my other question is similar

in an equilateral triangle ABC of sides 2x, a rectangle is constructed with the midpoints AB and AC as two vertices and the other two lying on bc. I understand one side is x but how do you find the breadth

Unfortunately, what you "understand" is not true. The angles in an equilateral triangle are 60 degrees. The legs of a right triangle with one angle 60 degrees and hypotenuse of length x/2 are x/4 and . The sides of the rectangle formed are x/2 and .

thnx in advance

posted by bloo,tomarto

Looking at this problem I am struck by the fact that many similar replies on this site use the Pythagorean Theorem instead of using the properties of standard triangles.
This one is a good example

radius of circle =x/2divided by radical 3
diameter of circle = x/radical 3 =diagonal of square
side of square =x/radical3radicical2
area= side squares=x^2/6


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