The perimeter of an isosceles triangle is 42 cm and its base is `1(1)/(2)` times each of the equal sides. Find (i) the length of each side of the triangle, (ii) the area of the triangle, and (iii) the height of the triangle. (Given, `sqrt(7) = 2.64`.)

Explanation

To solve the problem step by step, let's break it down into three parts as requested:

### Step 1: Finding the Length of Each Side of the Triangle

1. **Understanding the Problem**: We know that the perimeter of the isosceles triangle is 42 cm. The base of the triangle is \( \frac{3}{2} \) times the length of each of the equal sides.

2. **Let the Length of Each Equal Side be \( x \)**:
- The base will then be \( \frac{3}{2} x \).

3. **Setting Up the Perimeter Equation**:
- The perimeter \( P \) of the triangle is the sum of all its sides:
\[ P = x + x + \frac{3}{2}x = 42 \text{ cm} \]
- This simplifies to:
\[ 2x + \frac{3}{2}x = 42 \]

4. **Combining Like Terms**:
- Convert \( 2x \) to a fraction:
\[ 2x = \frac{4}{2}x \]
- Now combine:
\[ \frac{4}{2}x + \frac{3}{2}x = \frac{7}{2}x \]
- Set the equation:
\[ \frac{7}{2}x = 42 \]

5. **Solving for \( x \)**:
- Multiply both sides by \( \frac{2}{7} \):
\[ x = 42 \times \frac{2}{7} = 12 \text{ cm} \]

6. **Finding the Base**:
- Now calculate the base:
\[ \text{Base} = \frac{3}{2} \times 12 = 18 \text{ cm} \]

### Summary of Side Lengths:
- Each equal side: \( 12 \text{ cm} \)
- Base: \( 18 \text{ cm} \)

### Step 2: Finding the Area of the Triangle

1. **Using Heron's Formula**:
- First, calculate the semi-perimeter \( s \):
\[ s = \frac{12 + 12 + 18}{2} = 21 \text{ cm} \]

2. **Applying Heron's Formula**:
\[ \text{Area} = \sqrt{s(s-a)(s-b)(s-c)} \]
- Where \( a = 12 \), \( b = 12 \), and \( c = 18 \):
\[ \text{Area} = \sqrt{21(21-12)(21-12)(21-18)} = \sqrt{21 \times 9 \times 9 \times 3} \]

3. **Calculating the Area**:
- Simplify:
\[ = \sqrt{21 \times 81} = \sqrt{1701} \]
- Factor \( 1701 \):
\[ = 9 \sqrt{21} \]
- Given \( \sqrt{7} = 2.64 \), we can approximate \( \sqrt{21} \):
\[ \sqrt{21} = \sqrt{3 \times 7} = \sqrt{3} \times \sqrt{7} \approx 1.732 \times 2.64 \approx 4.57 \]
- Thus, the area is approximately:
\[ \text{Area} \approx 9 \times 4.57 \approx 41.13 \text{ cm}^2 \]

### Step 3: Finding the Height of the Triangle

1. **Using the Area Formula**:
- The area can also be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times \text{Base} \times \text{Height} \]
- We know the area and the base:
\[ 71.28 = \frac{1}{2} \times 18 \times h \]

2. **Solving for Height \( h \)**:
- Rearranging gives:
\[ h = \frac{71.28 \times 2}{18} = \frac{142.56}{18} \approx 7.92 \text{ cm} \]

### Final Results:
- (i) Length of each side: \( 12 \text{ cm}, 12 \text{ cm}, 18 \text{ cm} \)
- (ii) Area of the triangle: \( 71.28 \text{ cm}^2 \)
- (iii) Height of the triangle: \( 7.92 \text{ cm} \)