In an equilateral triangle with side 1, a square is inscribed so that one of the sides of the square lies

| September 6, 2021 | education

In an equilateral triangle with side 1, a square is inscribed so that one of the sides of the square lies on the side of the triangle, and on each of the other sides there is one vertex of the square. Find the side of the square.

Let’s build the height ВD of the equilateral triangle ABC.

Length ВD = АС * √3 / 2 = √3 / 2 cm.

The height BD of triangle ABC is also its median, then CD = AC / 2 = 1/2 cm.

Let the side of the CMHR square be X cm.

Then the length DH = X / 2, and the length of the segment CH = (1/2 – X / 2) = (1 – X) / 2 cm.

Right-angled triangles BDC and MNC are similar in acute angle, then:

BD / CD = MH / CH.

(√3 / 2) / (1/2) = X / ((1 – X) / 2).

√3 = 2 * X / (1 – X).

√3 – X * √3 = 2 * X.

2 * X + X * √3 = √3.

X * (2 + √3) = √3.

X = √3 / (2 + √3) cm.

Answer: The side of the square is √3 / (2 + √3) cm.



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