Question

The length of hypotenuse (in cm) of a right-angled triangle is 6 cm more than twice the length of its shortest side. If the length of its third side is 6 cm less than thrice the length of its shortest side, find the dimensions of the triangle.

Hint:

Check with Pythagoras triplet: \( 10^2 + 24^2 = 100 + 576 = 676 \), which is indeed \( 26^2 \).

Answer 1

Step 1: Let the Shortest Side be x
Shortest side = x
Other side = 3x − 6
Hypotenuse = 2x + 6

Step 2: Apply Pythagoras Theorem
Base² + Perpendicular² = Hypotenuse²

x² + (3x − 6)² = (2x + 6)²

Expand both sides:
x² + (9x² − 36x + 36) = 4x² + 24x + 36

Combine like terms:
10x² − 36x + 36 = 4x² + 24x + 36

Bring all terms to one side:
6x² − 60x = 0

Factorise:
6x(x − 10) = 0

x ≠ 0 (side cannot be zero)
x = 10

Step 3: Find Remaining Sides
Third side = 3x − 6
= 30 − 6
= 24 cm

Hypotenuse = 2x + 6
= 20 + 6
= 26 cm

Final Answer:
Sides are 10 cm, 24 cm, and 26 cm.