# Triangle ABC isosceles rectangular (angle BAC is 90), BC = 3cm. Point L lies on side AB, points E

Triangle ABC isosceles rectangular (angle BAC is 90), BC = 3cm. Point L lies on side AB, points E and K lie on side BC, and point D lies on AC so that LEKD is a square. Find the sides of the square LEKD.

1) The sum of all angles in a triangle is 180 degrees, then in a triangle ABC angle A + angle B + angle C = 180, angle C = 90, 180 – 90 = 90, angle A = angle B = 90: 2 = 45.

2) Consider a triangle DKS. In it, using the sum of the angles, we find the angle D.

Angle K = 90, because LDEK – square, angle C = 45, then angle D = 45 and the triangle is isosceles.

3) Similarly with the triangle BEL. Angle E = 90, angle B = 45, then angle L = 45 and the triangle is isosceles.

4) Because LDEK is a square, then all its sides are equal, i.e. LE = EK = DK and in turn are equal to the sides of the triangles KC and BE. Therefore, the BC side consists of three segments of equal length.

EK = BC: 3, EK = 3: 3 = 1 cm.

Answer: each side of the square is 1 cm.

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