The base of an isosceles triangle is 24 cm and its area is 192 sq. cm. Find its perimeter.

Answer

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

### Step 1: Understand the triangle
We have an isosceles triangle ABC, where AB = AC (the equal sides), and BC is the base which is given as 24 cm.

### Step 2: Use the area formula
The area of a triangle can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
Given that the area is 192 sq. cm and the base (BC) is 24 cm, we can set up the equation:
\[ 192 = \frac{1}{2} \times 24 \times h \]

### Step 3: Solve for height (h)
Rearranging the equation to find the height (h):
\[ 192 = 12h \quad \text{(since } \frac{1}{2} \times 24 = 12\text{)} \]
Now, divide both sides by 12:
\[ h = \frac{192}{12} = 16 \text{ cm} \]

### Step 4: Find the length of the equal sides (a)
Now, we can use the Pythagorean theorem in triangle ADC, where D is the midpoint of base BC. The length of DC is half of BC:
\[ DC = \frac{24}{2} = 12 \text{ cm} \]
Now, applying the Pythagorean theorem:
\[ a^2 = h^2 + DC^2 \]
Substituting the values we found:
\[ a^2 = 16^2 + 12^2 \]
Calculating the squares:
\[ a^2 = 256 + 144 = 400 \]
Now, taking the square root:
\[ a = \sqrt{400} = 20 \text{ cm} \]

### Step 5: Calculate the perimeter
The perimeter (P) of triangle ABC is given by:
\[ P = AB + AC + BC = a + a + 24 \]
Substituting the values:
\[ P = 20 + 20 + 24 = 64 \text{ cm} \]

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

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