60th Belarusian Mathematical Olympiad

Question

Given a right-angled triangle ABC with C = 90^. Its perimeter is equal to 30 cm. CH is the altitude of this triangle, CK and CL are the bisectors of the angles ACH and BCH respectively. Find the length of the hypotenuse AB if KL = 4 cm. FIGURE_0

Accepted Answer

Let a = BC, b = AC, c = AB, = BAC, = ABC, P = a+b+c. Then ACH = , BCH = , thus ACK = KCH = /2, HCL = LCB = /2. Since HKC = KAC + ACK (exterior angle of triangle ACK), we have HKC = + /2 = BCH + LCH = BCK. Hence triangle KCB is isosceles and BK = BC = a. Similarly, AL = AC = b. Therefore, AK = AB - BK = c - a, so KL = AL - AK = b - (c-a) = a+b-c. Hence c = (a+b+c) - (a+b-c)2 = P - KL2 = 30-42 = 13 (cm). FIGURE_0