In an isosceles right-angled triangle, each leg of which is equal to 2 meters, a square is inscribed with a common

| April 11, 2021 | education

In an isosceles right-angled triangle, each leg of which is equal to 2 meters, a square is inscribed with a common angle with it. Find the perimeter of the square.

Let a triangle ABC be given, the hypotenuse of which BC is equal (according to the Pythagorean theorem):

BC ^ 2 = AB ^ 2 + AC ^ 2 = 2 ^ 2 + 2 ^ 2 = 8 (m ^ 2).

Let the sides of the square AKDM be equal to x, AK = KD = DM = AM = x.

Points K, D, M – belong to the sides AB, BC and AC, respectively.

KB = MC = 2 – x, KD = MD = x.

Then, consider right-angled triangles KBD and MDC.

Their legs are equal to x and (2 – x), which means that their hypotenuses are equal.

Then, BC = 2 * BD.

By the Pythagorean theorem, BD ^ 2 = x ^ 2 + (2 – x) ^ 2 = x ^ 2 + 4 – 4 * x + x ^ 2 = 2 * x ^ 2 – 4 * x + 4 = 2 * (( x – 1) ^ 2 + 1).

BC ^ 2 = 8 = (2 * BD) ^ 2 = 4 * 2 * ((x – 1) ^ 2 + 1).

(x – 1) ^ 2 + 1 = 1.

(x – 1) ^ 2 = 0.

X = 1.

The perimeter of the square is P = 4 * x = 4 * 1 = 4 m.



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