The hypotenuse of a right-angled triangle is 3√5cm, and the difference between the legs is 3cm.

The hypotenuse of a right-angled triangle is 3√5cm, and the difference between the legs is 3cm. Find the legs and the perimeter of the right-angled triangle.

1. Let’s designate one leg as a, the second b.

2. According to the condition of the problem, one leg differs from the other by 3 cm, we write down the expression for their difference a – b = 3 cm.

3. To calculate the legs of a right-angled triangle, we use the Pythagorean theorem:

the square of the hypotenuse is equal to the sum of the squares of the legs.

(3 √5) ² = a² + b²; express a through b and get a = 3 + b.

Substitute into the equation

9 * 5 = (3 + b) ² + b² = 9 + 6 b + b² + b² = 2 b² + 6 b + 9 = 45;

Solve the quadratic equation and find only the positive root

b² + 3 b – 18 = 0;

b = (-3 + √ 9 + 4 * 18): 2 = (-3 + √81): 2 = (- 3 + 9): 2 = 3 cm.

The second leg is 3 + 3 = 6 cm.

4. Find the perimeter of the triangle

3 √5 cm + 6 cm + 3 cm = (3 √5 + 9) cm.