The base of an isosceles triangle is 48 cm and one of its equal sides is 30 cm. Find the area of the triangle 

Answer

To find the area of the isosceles triangle with a base of 48 cm and equal sides of 30 cm, we can follow these steps:

### Step 1: Identify the triangle and its components
We have an isosceles triangle ABC where:
- Base BC = 48 cm
- Equal sides AC = AB = 30 cm

### Step 2: Draw a perpendicular from A to BC
We draw a perpendicular line AD from point A to line BC. This perpendicular will bisect the base BC into two equal segments:
- BD = DC = 24 cm (since BC = 48 cm)

### Step 3: Use the Pythagorean theorem
In the right triangle ADC, we can apply the Pythagorean theorem:
\[ AC^2 = AD^2 + DC^2 \]
Where:
- AC = 30 cm (hypotenuse)
- DC = 24 cm (one leg)
- AD = height (the other leg)

### Step 4: Substitute the values into the Pythagorean theorem
Substituting the known values:
\[ 30^2 = AD^2 + 24^2 \]
\[ 900 = AD^2 + 576 \]

### Step 5: Solve for AD^2
Rearranging the equation to find AD^2:
\[ AD^2 = 900 - 576 \]
\[ AD^2 = 324 \]

### Step 6: Find the value of AD
Taking the square root of both sides:
\[ AD = \sqrt{324} \]
\[ AD = 18 \text{ cm} \]

### Step 7: Calculate the area of the triangle
The area \( A \) of triangle ABC can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Substituting the values:
\[ A = \frac{1}{2} \times 48 \times 18 \]

### Step 8: Simplify the area calculation
Calculating the area:
\[ A = \frac{1}{2} \times 48 \times 18 = 24 \times 18 = 432 \text{ cm}^2 \]

### Final Answer
The area of the triangle ABC is **432 cm²**.

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