The base of a right triangle is 48 cm and its hypotenuse is 50 cm long. The area of the triangle is

Options

A. 168 `cm^(2)`
B. 252 `cm^(2)`
C. 336 `cm^(2)`
D. 504 `cm^(2)`

Answer

To find the area of the right triangle with a base of 48 cm and a hypotenuse of 50 cm, we can follow these steps:

### Step 1: Identify the triangle and its components
We have a right triangle ABC where:
- Base (BC) = 48 cm
- Hypotenuse (AC) = 50 cm
- We need to find the height (AB) to calculate the area.

### Step 2: Use the Pythagorean theorem
In a right triangle, the Pythagorean theorem states that:
\[ \text{Hypotenuse}^2 = \text{Base}^2 + \text{Height}^2 \]
Let the height AB be represented as \( x \) cm. Thus, we can write:
\[ 50^2 = 48^2 + x^2 \]

### Step 3: Calculate the squares
Now, calculate the squares:
- \( 50^2 = 2500 \)
- \( 48^2 = 2304 \)

### Step 4: Substitute and solve for \( x \)
Substituting the values into the equation:
\[ 2500 = 2304 + x^2 \]
Now, isolate \( x^2 \):
\[ x^2 = 2500 - 2304 \]
\[ x^2 = 196 \]

### Step 5: Find the value of \( x \)
Taking the square root of both sides:
\[ x = \sqrt{196} \]
\[ x = 14 \text{ cm} \]
So, the height AB is 14 cm.

### Step 6: Calculate the area of the triangle
The formula for the area of a triangle is:
\[ \text{Area} = \frac{1}{2} \times \text{Base} \times \text{Height} \]
Substituting the values we have:
\[ \text{Area} = \frac{1}{2} \times 48 \times 14 \]

### Step 7: Perform the multiplication
Calculating the multiplication:
\[ 48 \times 14 = 672 \]

### Step 8: Final calculation for area
Now, calculate the area:
\[ \text{Area} = \frac{1}{2} \times 672 = 336 \text{ cm}^2 \]

### Conclusion
The area of the triangle is \( 336 \text{ cm}^2 \).

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