The area of a right angled triangle is 24 cm and one of the sides containing the right angle is 6 cm. The altitude on the hypotenuse is

Options

A. 3.6 cm
B. 4.8 cm
C. 5.2 cm
D. 12 cm

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Understand the given information
We know that:
- The area of the right-angled triangle = 24 cm²
- One of the sides containing the right angle (base) = 6 cm

### Step 2: Use the area formula for a triangle
The formula for the area of a triangle is given by:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, we can consider the base as 6 cm and let the height be \( h \).

### Step 3: Set up the equation
Substituting the known values into the area formula:
\[ 24 = \frac{1}{2} \times 6 \times h \]

### Step 4: Simplify the equation
Multiply both sides by 2 to eliminate the fraction:
\[ 48 = 6 \times h \]

### Step 5: Solve for height
Now, divide both sides by 6 to find \( h \):
\[ h = \frac{48}{6} = 8 \text{ cm} \]

### Step 6: Find the hypotenuse using Pythagorean theorem
Now that we have both the base (6 cm) and the height (8 cm), we can find the hypotenuse \( c \) using the Pythagorean theorem:
\[ c^2 = \text{base}^2 + \text{height}^2 \]
Substituting the values:
\[ c^2 = 6^2 + 8^2 = 36 + 64 = 100 \]
Taking the square root:
\[ c = \sqrt{100} = 10 \text{ cm} \]

### Step 7: Calculate the altitude on the hypotenuse
Now, we need to find the altitude \( h_a \) from the right angle to the hypotenuse. The area can also be expressed using the hypotenuse as the base:
\[ \text{Area} = \frac{1}{2} \times \text{hypotenuse} \times \text{altitude} \]
Substituting the known values:
\[ 24 = \frac{1}{2} \times 10 \times h_a \]

### Step 8: Simplify and solve for altitude
Multiply both sides by 2:
\[ 48 = 10 \times h_a \]
Now, divide by 10:
\[ h_a = \frac{48}{10} = 4.8 \text{ cm} \]

### Final Answer
The altitude on the hypotenuse is **4.8 cm**.

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