Find the area of a right triangle having base = 24 cm and hypotenuse = 25 cm.

Options

A. 48 `cm^2`
B. 54 `cm^2`
C. 64 `cm^2`
D. 84 `cm^2`

Answer

To find the area of a right triangle with a base of 24 cm and a hypotenuse of 25 cm, we can follow these steps:

### Step 1: Identify the given values
- Base (b) = 24 cm
- Hypotenuse (h) = 25 cm

### Step 2: Use the Pythagorean theorem
In a right triangle, the Pythagorean theorem states that:
\[ h^2 = b^2 + p^2 \]
where \( p \) is the perpendicular (height) of the triangle.

### Step 3: Substitute the known values into the Pythagorean theorem
\[ 25^2 = 24^2 + p^2 \]

### Step 4: Calculate the squares
- \( 25^2 = 625 \)
- \( 24^2 = 576 \)

Now substitute these values into the equation:
\[ 625 = 576 + p^2 \]

### Step 5: Solve for \( p^2 \)
Rearranging the equation gives:
\[ p^2 = 625 - 576 \]
\[ p^2 = 49 \]

### Step 6: Find the value of \( p \)
Taking the square root of both sides:
\[ p = \sqrt{49} \]
\[ p = 7 \text{ cm} \]

### Step 7: Calculate the area of the triangle
The area \( A \) of a right triangle is given by the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Substituting the values we have:
\[ A = \frac{1}{2} \times 24 \times 7 \]

### Step 8: Perform the multiplication
\[ A = \frac{1}{2} \times 168 \]
\[ A = 84 \text{ cm}^2 \]

### Final Answer
The area of the right triangle is \( 84 \text{ cm}^2 \).

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