The perimeter of a rectangle is 46 m and its length is 15 m. Find its :
(i) breadth
(ii) area
(iii) diagonal

Answer

To solve the problem step by step, let's break it down into three parts: finding the breadth, the area, and the diagonal of the rectangle.

### Given:
- Perimeter of the rectangle (P) = 46 m
- Length of the rectangle (l) = 15 m

### (i) Finding the Breadth

1. **Use the formula for the perimeter of a rectangle:**
\[ P = 2(l + b) \]
where \( b \) is the breadth.

2. **Substitute the known values into the formula:**
\[ 46 = 2(15 + b) \]

3. **Divide both sides by 2:**
\[ 23 = 15 + b \]

4. **Isolate \( b \) by subtracting 15 from both sides:**
\[ b = 23 - 15 \]
\[ b = 8 \, \text{m} \]

### (ii) Finding the Area

1. **Use the formula for the area of a rectangle:**
\[ \text{Area} = l \times b \]

2. **Substitute the values of length and breadth:**
\[ \text{Area} = 15 \times 8 \]

3. **Calculate the area:**
\[ \text{Area} = 120 \, \text{m}^2 \]

### (iii) Finding the Diagonal

1. **Use the Pythagorean theorem to find the diagonal \( d \):**
\[ d^2 = l^2 + b^2 \]

2. **Substitute the values of length and breadth:**
\[ d^2 = 15^2 + 8^2 \]

3. **Calculate \( 15^2 \) and \( 8^2 \):**
\[ d^2 = 225 + 64 \]

4. **Add the squares:**
\[ d^2 = 289 \]

5. **Take the square root to find \( d \):**
\[ d = \sqrt{289} = 17 \, \text{m} \]

### Summary of Results:
- (i) Breadth = 8 m
- (ii) Area = 120 m²
- (iii) Diagonal = 17 m

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