A square is inscribed in a right-angled isosceles triangle, so that one side lies on the hypotenuse.

| May 23, 2021 | education

A square is inscribed in a right-angled isosceles triangle, so that one side lies on the hypotenuse. Find the perimeter of a square?

Triangle ABC is isosceles and rectangular, therefore, its angles BAC and BCA are equal to 45.

Consider a triangle АDН, in which the angle DHA is straight, since the side of the square lies on the hypotenuse AC, the angle DАН = 45, therefore, the angle АDН = 45. Then the triangle АDН is isosceles, it has DH = АН.

Consider a CMР triangle, in which the CMР angle is straight, since the side of the square lies on the hypotenuse AC, the CMР angle = 45, therefore, the CCM angle = 45. Then the CMР triangle is isosceles, its CM = CM.

The НM segment on the hypotenuse is equal to the length of the side of the square DKMН.

Thus, the side of the square divides the hypotenuse into three equal parts, which are equal to the side of the square.

Then Psquare = AC + AC / 3 = AC * 4/3.

Answer: The perimeter of the square is AC * 4/3.



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