The base of an isosceles triangle is 24 cm and its area is `60 cm^(2) `Find its perimeter.

Answer

To find the perimeter of the isosceles triangle with a base of 24 cm and an area of 60 cm², we can follow these steps:

### Step 1: Understand the triangle
We have an isosceles triangle ABC with base BC = 24 cm. The area of the triangle is given as 60 cm².

### Step 2: Use the area formula
The area (A) of a triangle can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, the base (BC) = 24 cm and the area = 60 cm².

### Step 3: Set up the equation
Substituting the known values into the area formula:
\[ 60 = \frac{1}{2} \times 24 \times h \]
where \( h \) is the height from point A to the base BC.

### Step 4: Solve for height (h)
To isolate \( h \), we can rearrange the equation:
\[ 60 = 12h \quad \Rightarrow \quad h = \frac{60}{12} = 5 \text{ cm} \]
So, the height (AD) from point A to the base BC is 5 cm.

### Step 5: Find the lengths of the equal sides (AC and AB)
Since D is the midpoint of BC, we have:
\[ BD = DC = \frac{24}{2} = 12 \text{ cm} \]
Now, we can use the Pythagorean theorem in triangle ABD:
\[ AB^2 = AD^2 + BD^2 \]
Substituting the known values:
\[ AB^2 = 5^2 + 12^2 = 25 + 144 = 169 \]
Taking the square root gives:
\[ AB = \sqrt{169} = 13 \text{ cm} \]
Since the triangle is isosceles, AC = AB = 13 cm.

### Step 6: Calculate the perimeter
The perimeter (P) of triangle ABC is given by:
\[ P = AB + BC + AC \]
Substituting the values we found:
\[ P = 13 + 24 + 13 = 50 \text{ cm} \]

### Final Answer
The perimeter of the triangle is **50 cm**.

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