Question

The sides of a right-angled triangle are 2x, x + 5 and 3x + 1. If the hypotenuse is 3x + 1, find the sides of triangle.

Answer 1

Given:

Sides: 2x, x + 5, 3x + 1 hypotenuse = 3x + 1.

Use Pythagoras’ theorem for a right-angled triangle.

Step-wise calculation:

1. Apply Pythagoras:

(2x)² + (x + 5)² 

= (3x + 1)²

2. Expand:

4x² + (x² + 10x + 25)

= 9x² + 6x + 1

3. Combine terms:

5x² + 10x + 25

= 9x² + 6x + 1

4. Bring all to one side:

= 9x² + 6x + 1 – 5x² – 10x – 25

= 4x² – 4x – 24

5. Divide by 4:

x² – x – 6 = 0

6. Factor:

(x – 3)(x + 2) = 0

⇒ x = 3 or x = –2

7. Discard x = –2, side lengths must be positive. 

So, x = 3.

Sides are 2x = 6, x + 5 = 8 and hypotenuse 3x + 1 = 10.

The triangle sides are 6, 8, 10.