The sum of the lengths of the hypotenuse and leg of a right-angled triangle is 9 cm, and their difference is 4 cm. Find another leg.

1) Find the length of the first leg and hypotenuse.

Let the length of the leg be x cm, then the length of the hypotenuse is (9 – x) cm.By the condition of the problem, it is known that the difference between the hypotenuse and the leg is ((9 – x) – x) cm or 4 cm.Let’s make an equation and solve it.

(9 – x) – x = 4;

9 – x – x = 4;

9 – 2x = 4;

-2x = 4 – 9;

-2x = -5;

x = -5: (-2);

x = 2.5 (cm) – the length of the first leg;

9 – x = 9 – 2.5 = 6.5 (cm) – length of the hypotenuse.

2) Find the length of the second leg. By the Pythagorean theorem: The square of the hypotenuse is equal to the sum of the squares of the legs, we have:

6.5 ^ 2 = x ^ 2 + 2.5 ^ 2;

x ^ 2 = 6.5 ^ 2 – 2.5 ^ 2;

x ^ 2 = 42.25 – 6.25;

x ^ 2 = 36;

x = 6 (cm) – the length of the second leg.

Answer. 6 cm.