The hypotenuse of a right-angled triangle is 13 cm. If one of its legs is increased by 4 cm

The hypotenuse of a right-angled triangle is 13 cm. If one of its legs is increased by 4 cm, then the hypotenuse will increase by 2 cm. Find the legs of the triangle.

In a right triangle:

1. The hypotenuse is equal to: Lg = 13 cm;

2. The first leg is equal to: Lк1 cm;

3. The second leg is equal to: Lk2 cm;

4. By the Pythagorean theorem:

Lg ^ 2 = Lk1 ^ 2 + Lk2 ^ 2;

4. Let’s increase the first leg by: Lo = 4 cm;

5. The hypotenuse will lengthen by: Lgo = 2 cm;

Lk11 = Lk1 + Lo = (Lk1 + 4) cm;

Lgo = Lg + Lgo = (Lg + 2) cm;

6. Let’s calculate the new value of the hypotenuse:

Lg11 ^ 2 = Lk11 ^ 2 + Lk2 ^ 2;

(Lg + 2) ^ 2 = (Lk1 + 4) ^ 2 + Lk2 ^ 2;

Lg ^ 2 + 4 * Lg + 4 = Lk1 ^ 2 + 8 * Lk1 + 16 + Lk2 ^ 2;

(Lg ^ 2 – (Lk1 ^ 2 + Lk2 ^ 2)) + 4 * Lg = 8 * Lk1 + 16 – 4;

4 * Lg = 8 * 8 * Lk1 + 12;

8 * Lk1 = 4 * Lg – 12 = 4 * 13 – 12 = 52 – 12 = 40 cm;

Lк1 = 40/8 = 5 cm;

7. Second leg:

Lk2 = √ (Lg ^ 2 – Lk1 ^ 2) = √ (13 ^ 2 – 5 ^ 2) = 12 cm.

Answer: the length of the first leg is 5 cm, the second is 12 cm.