Question

Find the area of a right angle triangle with hypotenuse 25 cm and one side 24 cm

• A. 54 cm$^2$
• B. 84 cm$^2$
• C. 96 cm$^2$
• D. 124 cm$^2$

Hint:

Recognizing Pythagorean triplets can save valuable exam time. The triplet $(7, 24, 25)$ is a standard set. If the hypotenuse is 25 and one side is 24, the other side is immediately 7.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
A right-angled triangle contains one angle measuring $90^\circ$.
The side opposite to the right angle is the hypotenuse, which is the longest side.
The relationship between the sides is governed by the Pythagorean Theorem.
Key Formula or Approach:
According to the Pythagorean Theorem: \[ c^2 = a^2 + b^2 \] where:
- $c$ is the hypotenuse.
- $a$ and $b$ are the other two perpendicular sides.
The area of a right-angled triangle is calculated using: \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times a \times b \]

Step 2: Detailed Explanation:
Let the given values be:
- Hypotenuse ($c$) = $25$ cm
- One side ($a$) = $24$ cm
We need to determine the length of the unknown side ($b$): \[ 25^2 = 24^2 + b^2 \] Calculate the squares of the numbers: \[ 625 = 576 + b^2 \] Isolate $b^2$: \[ b^2 = 625 - 576 \] \[ b^2 = 49 \] Find the square root: \[ b = \sqrt{49} = 7 \text{ cm} \] Now that we have both perpendicular sides ($24$ cm and $7$ cm), calculate the area: \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \] \[ \text{Area} = \frac{1}{2} \times 24 \times 7 \] \[ \text{Area} = 12 \times 7 = 84 \text{ cm}^2 \]

Step 3: Final Answer:
The area of the right-angled triangle is 84 cm$^2$.