Question

Which of the following can be the sides of a right angle triangle?

• A. 5 cm, 13 cm, 12 cm
• B. 4 cm, 4 cm, 10 cm
• C. 3 cm, 3 cm, 5 cm
• D. 6 cm, 6 cm, 10 cm

Hint:

- Pythagorean Theorem: In a right-angled triangle with legs \(a, b\) and hypotenuse \(c\), \(a^2 + b^2 = c^2\). - The hypotenuse is always the longest side. - Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle must be greater than the length of the third side. (e.g., for option (2), \(4+4=8\), which is not greater than 10, so it doesn't form any triangle). - Common Pythagorean triples: (3,4,5), (5,12,13), (8,15,17), (7,24,25).

Answer 1

The Correct Option is A

Step 1: Recall the Pythagorean theorem for a right-angled triangle.
In a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides (legs).
If \(a, b\) are the lengths of the legs and \(c\) is the length of the hypotenuse, then \( a^2 + b^2 = c^2 \).
The hypotenuse is always the longest side.

Step 2: Check each option.
Option (1) Sides: 5 cm, 12 cm, 13 cm.
Longest side is 13 cm (potential hypotenuse).
Check: \( 5^2 + 12^2 = 25 + 144 = 169 \).
Hypotenuse squared: \( 13^2 = 169 \).
Since \( 5^2 + 12^2 = 13^2 \) (i.
e.
, \(169 = 169\)), these sides can form a right-angled triangle.
This is a Pythagorean triple (5, 12, 13).
This statement is TRUE.
Option (2) Sides: 4 cm, 4 cm, 10 cm.
Longest side is 10 cm.
Check: \( 4^2 + 4^2 = 16 + 16 = 32 \).
Hypotenuse squared: \( 10^2 = 100 \).
Since \( 32 \ne 100 \), these sides cannot form a right-angled triangle.
(Also, triangle inequality: \(4+4=8 \not> 10\), so these can't even form a triangle).
Option (3) Sides: 3 cm, 3 cm, 5 cm.
Longest side is 5 cm.
Check: \( 3^2 + 3^2 = 9 + 9 = 18 \).
Hypotenuse squared: \( 5^2 = 25 \).
Since \( 18 \ne 25 \), these sides cannot form a right-angled triangle.
Option (4) Sides: 6 cm, 6 cm, 10 cm.
Longest side is 10 cm.
Check: \( 6^2 + 6^2 = 36 + 36 = 72 \).
Hypotenuse squared: \( 10^2 = 100 \).
Since \( 72 \ne 100 \), these sides cannot form a right-angled triangle.

Step 3: Identify the set of sides that forms a right-angled triangle.
Option (1) satisfies the Pythagorean theorem.
This matches option (1).