The base of an isosceles triangle measures 24 cm and its area is 192 `cm^(2)` ? Find its perimeter.

Options

A. 64cm
B. 46cm
C. 84cm
D. 54cm

Answer

To find the perimeter of the isosceles triangle with a base of 24 cm and an area of 192 cm², we can follow these steps:

### Step 1: Understand the problem and gather the information.
We know:
- Base (b) = 24 cm
- Area (A) = 192 cm²
- The triangle is isosceles, meaning two sides are equal.

### Step 2: Use the formula for the area of a triangle.
The area of a triangle can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Let the height be \( h \). Thus, we can write:
\[ 192 = \frac{1}{2} \times 24 \times h \]

### Step 3: Solve for the height (h).
Rearranging the equation gives:
\[ 192 = 12h \implies h = \frac{192}{12} = 16 \text{ cm} \]

### Step 4: Find the length of the equal sides (a).
In an isosceles triangle, we can use the Pythagorean theorem to find the length of the equal sides. The height divides the base into two equal parts, so each part is:
\[ \frac{24}{2} = 12 \text{ cm} \]
Now, using the Pythagorean theorem:
\[ a^2 = h^2 + \left(\frac{b}{2}\right)^2 \]
Substituting the known values:
\[ a^2 = 16^2 + 12^2 \]
Calculating:
\[ a^2 = 256 + 144 = 400 \implies a = \sqrt{400} = 20 \text{ cm} \]

### Step 5: Calculate the perimeter (P).
The perimeter of the triangle is the sum of all its sides:
\[ P = a + a + b = 20 + 20 + 24 = 64 \text{ cm} \]

### Final Answer:
The perimeter of the triangle is **64 cm**.

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