In triangle ABC the angle C is equal to 90º. On the sides АС, AB, ВС the points М, Р, К

In triangle ABC the angle C is equal to 90º. On the sides АС, AB, ВС the points М, Р, К are respectively taken so that the quadrangle СМРК is a square, АС = 6 cm, ВС = 14 cm. Find the side of the MC.

Let’s denote the length of the side of the square through x:

MC = MP = KP = CK = x.

The APM triangle is similar to the BKP triangle because they have a right angle and <APM = <ABK (one-sided corresponding when cutting parallel lines CB and MP).

KB / MP = KP / MA (1);

KB = BC – CK = BC – x;

KP = MC;

AM = AC – MC = AC – x;

Substitution in (1) expressions for KB, KP, AM, MP:

(BC – x) / x = x / (AC – x);

(14 – x) / x = x / (6 – x);

(14 – x) (6 – x) = x ^ 2;

84 – 14x – 6x + x ^ 2 = x ^ 2;

20x = 84;

x = 4.2.

Answer: MC = 4.2.