The perimeter of an isosceles triangle is 42 cm and its base is `1(1)/(2)` times each of the equal sides. Find the area of triangle

Options

A. `71.43 cm^(2)`
B. `60.45 cm^(2)`
C. `70.50 cm^(2)`
D. `78.73 cm^(2)`

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Understand the Problem
We have an isosceles triangle with a perimeter of 42 cm. The base is \( \frac{3}{2} \) times the length of each of the equal sides.

### Step 2: Define Variables
Let the length of each of the equal sides be \( x \). Then, the base of the triangle can be expressed as:
\[ \text{Base} = \frac{3}{2} x \]

### Step 3: Write the Perimeter Equation
The perimeter of the triangle is the sum of all its sides:
\[ x + x + \frac{3}{2} x = 42 \]
This simplifies to:
\[ 2x + \frac{3}{2}x = 42 \]

### Step 4: Combine Like Terms
To combine the terms, convert \( 2x \) into a fraction:
\[ 2x = \frac{4}{2}x \]
Thus, the equation becomes:
\[ \frac{4}{2}x + \frac{3}{2}x = 42 \]
Combine the fractions:
\[ \frac{7}{2}x = 42 \]

### Step 5: Solve for \( x \)
To isolate \( x \), multiply both sides by \( \frac{2}{7} \):
\[ x = 42 \times \frac{2}{7} = 12 \]

### Step 6: Find the Base Length
Now, substitute \( x \) back to find the base:
\[ \text{Base} = \frac{3}{2} \times 12 = 18 \text{ cm} \]

### Step 7: Find the Height
To find the height \( h \), we can use the Pythagorean theorem. The height divides the base into two equal segments of \( \frac{18}{2} = 9 \) cm. The height \( h \) can be calculated as follows:
\[ h = \sqrt{x^2 - \left(\frac{\text{Base}}{2}\right)^2} = \sqrt{12^2 - 9^2} \]
Calculating the squares:
\[ h = \sqrt{144 - 81} = \sqrt{63} \]

### Step 8: Calculate the Area
The area \( A \) of the triangle can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{Base} \times \text{Height} \]
Substituting the values:
\[ A = \frac{1}{2} \times 18 \times \sqrt{63} \]
Calculating:
\[ A = 9 \sqrt{63} \text{ cm}^2 \]

### Final Answer
The area of the triangle is:
\[ A = 9 \sqrt{63} \text{ cm}^2 \]