Which of the following can be the possible lengths (in cm) of a right angled triangle?

Question

Accepted Answer

The correct option is 2:
To determine which set of lengths can form a right-angled triangle, we need to recall and apply the Pythagorean theorem.

**Key Concept:** The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle, which is always the longest side) is equal to the sum of the squares of the lengths of the other two sides (legs). Mathematically, if $a$ and $b$ are the lengths of the legs and $c$ is the length of the hypotenuse, then:
$$a^2 + b^2 = c^2$$

We will test each option by taking the two shorter lengths as $a$ and $b$, and the longest length as $c$.

Let's examine Option 2: **7, 24, 25**
Here, the two shorter sides are $a = 7$ cm and $b = 24$ cm, and the longest side (hypotenuse) is $c = 25$ cm.
Substitute these values into the Pythagorean theorem:
$a^2 + b^2 = 7^2 + 24^2$
$7^2 = 49$
$24^2 = 576$
$49 + 576 = 625$

Now, calculate the square of the hypotenuse:
$c^2 = 25^2 = 625$

Since $a^2 + b^2 = c^2$ ($625 = 625$), this set of lengths satisfies the Pythagorean theorem. Therefore, 7 cm, 24 cm, and 25 cm can be the lengths of a right-angled triangle.

The incorrect option is 1:
The option values are 8, 10, 15. If we apply the Pythagorean theorem ($8^2 + 10^2 = 64 + 100 = 164$), it does not equal $15^2 = 225$. So, this cannot be a right-angled triangle.

The incorrect option is 3:
The option values are 3, 5, 6. If we apply the Pythagorean theorem ($3^2 + 5^2 = 9 + 25 = 34$), it does not equal $6^2 = 36$. So, this cannot be a right-angled triangle.

The incorrect option is 4:
The option values are 18, 10, 12. First, order the sides: 10, 12, 18. If we apply the Pythagorean theorem ($10^2 + 12^2 = 100 + 144 = 244$), it does not equal $18^2 = 324$. So, this cannot be a right-angled triangle.