Question

In ΔАВC, ∠B = 90°. If AC = (x + 4) cm, BC = (x + 2) cm and AB = (3x + 1) cm, find the sides of triangle.

Answer 1

Given: In triangle ABC

∠B = 90°

AC = (x + 4) cm

BC = (x + 2) cm

AB = (3x + 1) cm

Step-wise calculation:

1. Use Pythagoras since ∠B = 90°: 

AB² + BC² = AC²

2. Substitute expressions: 

(3x + 1)² + (x + 2)² = (x + 4)²

3. Expand: 

9x² + 6x + 1 + x² + 4x + 4

= x² + 8x + 16

4. Combine like terms and move to one side: 

10x² + 10x + 5 – x² – 8x – 16 = 0

⇒ 9x² + 2x – 11 = 0

5. Solve the quadratic: 

Discriminant D 

= 2² – 4 × 9 × (–11) 

= 4 + 396

= 400

`sqrt(D) = 20xx x`

= `(-2 ± 20)/(18)` 

⇒ x = 1 or x = `-11/9`.

6. Check validity: Side lengths must be positive.

For x = 1:

AC = 5 cm

BC = 3 cm

AB = 4 cm, all positive.

For x = `-11/9`: 

AB = 3x + 1

= `-8/3` (negative) reject.

The triangle’s sides are AB = 4 cm, BC = 3 cm and AC = 5 cm (a 3–4–5 right triangle).