In $\bigtriangleup$ XYZ, /XY/ = 8cm, /YZ/ = 10cm and /XZ/ = 6cm. Which of these relation is true?

Analyze the question: We are given the three sides of triangle XYZ: XY = 8 cm, YZ = 10 cm, XZ = 6 cm. We need to find which relation is true.

Concept: Check whether the sides satisfy the Pythagorean theorem. In a right-angled triangle, the square of the longest side (hypotenuse) equals the sum of the squares of the other two sides.

Solution:

The longest side is YZ = 10 cm.

• YZ² = 10² = 100
• XZ² + XY² = 6² + 8² = 36 + 64 = 100

So YZ² = XZ² + XY², which means the triangle is right-angled at X.

Checking the other options:

• A: 8 + 10 = 18 ≠ 6, so it is false.
• B: 8 − 10 = −2 ≠ 6, so it is false.
• C: The option as written, /XY/² = /Y/² − /XY/², is malformed (it has /Y/ instead of a side length). Even read as XY² = YZ² − XY², we get 64 = 100 − 64 = 36, which is false.
• D: YZ² = XZ² + XY² gives 100 = 100, which is true.

Answer: D.

Tip: Whenever you see the numbers 6, 8, 10 (or 3, 4, 5, and 5, 12, 13), think "Pythagorean triple." The longest side is always the one that stands alone on one side of the equation.