The base of a right prism is a right-angled triangle whose sides are 5 cm, 12 cm and 13 cm. If the area of the total surface the prism is `360 cm^(2)`, then its height (in cm) is

Options

A. 10
B. 12
C. 9
D. 11

Answer

To find the height of the right prism, we will follow these steps:

### Step 1: Calculate the area of the base triangle
The base of the prism is a right-angled triangle with sides 5 cm, 12 cm, and 13 cm. The area \( A \) of a right-angled triangle can be calculated using the formula:

\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]

Here, we can take the base as 5 cm and the height as 12 cm.

\[ A = \frac{1}{2} \times 5 \times 12 = \frac{1}{2} \times 60 = 30 \, \text{cm}^2 \]

### Step 2: Calculate the perimeter of the base triangle
The perimeter \( P \) of the triangle is the sum of all its sides:

\[ P = 5 + 12 + 13 = 30 \, \text{cm} \]

### Step 3: Write the formula for the total surface area of the prism
The total surface area \( TSA \) of a prism is given by the formula:

\[ TSA = 2 \times \text{Base Area} + \text{Perimeter} \times \text{Height} \]

Substituting the values we found:

\[ TSA = 2 \times 30 + 30 \times h \]

### Step 4: Set up the equation using the given total surface area
We know the total surface area is given as 360 cm²:

\[ 360 = 2 \times 30 + 30h \]

### Step 5: Simplify the equation
Calculating the left side:

\[ 360 = 60 + 30h \]

### Step 6: Isolate the height \( h \)
Subtract 60 from both sides:

\[ 360 - 60 = 30h \]
\[ 300 = 30h \]

Now, divide both sides by 30:

\[ h = \frac{300}{30} = 10 \, \text{cm} \]

### Final Answer
The height of the prism is \( 10 \, \text{cm} \).

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