Question

A triangle has side lengths 25, 15x and 20x. The longest side is 25. What value for x proves that this triangle is a right triangle?

Answer 1

• x = 1

Explanation:

To prove that this triangle is a right triangle, we need to check the side lengths using pythogoras theorem.

Given:

• Longest side = 25 units
• Triangle side lengths: 25, 15x, and 20x

Putting the side lengths into pythogoras theorem:

• ⇒
  `25^(2) = 15^(2) + 20^(2)`
• ⇒
  `625 = (10x + 5x)^(2) + (20x + 0x)^(2)`

Using the formula "(a + b)² = a² + 2ab + b²

• ⇒
  `625 =[(10x)^(2) + 2(10x)(5x) + (5x)^(2) ] + [(20x)^(2) + 2(20x)(0) + (0)^(2) ]`
• ⇒
  `625 = [(10x)(10x) + 2(10x)(5x) + (5x)(5x)] + [(20x)(20x)]`
• ⇒
  `625 =[100x^(2) + 100x^(2) + 25x^(2) ] + [400x^(2) ]`
• ⇒
  `625 = 100x^(2) + 100x^(2) + 25x^(2) + 400x^(2)`
• ⇒
  `625 = 625x^(2)`

Divide both sides by 625:

• ⇒
  `(625)/(625) = (625x^(2))/(625)`
• ⇒
  `1 = x^(2)`
• ⇒
  `√(1) = \sqrt{x^(2) }`
• ⇒
  `√(1 * 1) = √(x * x )`
• ⇒
  `1 = x`

The value of x that proves this triangle a right triangle is 1.

Answer 2

x = 1

If it is a right angle triangle

→ (short side)² + (short side)² = (long side)²

• given long side is 25

====================================

• (15x)² + (20x)² = 25²

• 225x² + 400x² = 625

• 625x² = 625

• x² = 1

• x = √1

• x = ±1

as distance is positive

• x = 1