The leg of a right-angled triangle is 12 cm.Find another leg and hypotenuse if they are proportional to the numbers 3 and 5.

Let’s denote the legs of the triangle a and b, with a = 12 cm, and the hypotenuse is equal to c. Moreover, it is known that in: c = 3: 5. Let’s denote the values in and c through the coefficient k, then in = 3 * k; c = 5 * k. Apply to sides a, b and c the theorem expressed by the formula:

a ^ 2 + b ^ 2 = c ^ 2; substitute the ratios according to the condition, we get: 12 ^ 2 + (3 * k) ^ 2 = (5 * k) ^ 2; 144 + (9 * k) ^ 2 = 25 * k ^ 2; 144 = (25 – 9) * k ^ 2; 144 = 16 * k ^ 2; k ^ 2 = 144: 16 = 9.

Whence k = √9 = 3. Let’s define in and with: in = 3 * k = 3 * 3 = 9 cm; c = 5 * k = 3 * 5 = 15. a ^ 2 + b ^ 2 = c ^ 2; 12 ^ 2 + 9 ^ 2 = 15 ^ 2; 144 + 81 = 225.