Through the point L lying on the hypotenuse AB of an isosceles right triangle ABC, segments LK and LM

Through the point L lying on the hypotenuse AB of an isosceles right triangle ABC, segments LK and LM are drawn perpendicular to the sides AC and BC, respectively. Find the leg of the triangle if the perimeter of the quadrilateral LKCM is 12cm

Triangle ABC is rectangular with right angle C and isosceles, therefore angles CAB and CBA are equal to 45. Consider a triangle LMB, in which the angle M is right, and angles B and L are 45, hence this triangle is isosceles and LM = MB.

It can be seen from the figure that the leg CB of the triangle ABC is equal to the sum of CM + MV, but we found out that MV = LM, and CM = KL, since these are opposite sides of the rectangle, then the leg CB = KL + LM.

Perimeter of the KCML rectangle P = 2 * KL + 2 * ML = 12.

KL + ML = 6 = CB.

Answer: The leg of the triangle ABC = 6 cm.