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: Understanding the Concept:
Use Pythagoras Theorem: \( \text{Base}^2 + \text{Perpendicular}^2 = \text{Hypotenuse}^2 \). Define all sides in terms of the shortest side.
Step 2: Detailed Explanation:
Let shortest side be \( x \).
Hypotenuse \( = 2x + 6 \).
Third side \( = 3x - 6 \).
By Pythagoras Theorem:
\[ x^2 + (3x - 6)^2 = (2x + 6)^2 \]
\[ x^2 + (9x^2 - 36x + 36) = (4x^2 + 24x + 36) \]
\[ 10x^2 - 36x + 36 = 4x^2 + 24x + 36 \]
\[ 6x^2 - 60x = 0 \implies 6x(x - 10) = 0 \]
Since side length cannot be 0, \( x = 10 \).
Shortest side \( = 10 \) cm.
Third side \( = 3(10) - 6 = 24 \) cm.
Hypotenuse \( = 2(10) + 6 = 26 \) cm.
Step 3: Final Answer:
Dimensions are 10 cm, 24 cm, and 26 cm.