ABCD is a parallelogram whose adjacent sides AB = 48 cm and BC = 14 cm . One of its diagonds AC = 50 cm . Find :
(i) the area of the parallelogram ABCD.
(ii) the distance between the longer sides.
(iii) the distance between the shorter sides.

Answer

To solve the problem step by step, we will find the area of the parallelogram ABCD, the distance between the longer sides, and the distance between the shorter sides.

### Step 1: Find the area of the parallelogram ABCD

1. **Identify the sides and diagonal**:
- AB = 48 cm
- BC = 14 cm
- AC (diagonal) = 50 cm

2. **Calculate the semi-perimeter (s) of triangle ABC**:
\[ s = \frac{AB + BC + AC}{2} = \frac{48 + 14 + 50}{2} = \frac{112}{2} = 56 \text{ cm} \]

3. **Use Heron's formula to find the area of triangle ABC**:
- Area \(A\) of triangle ABC can be calculated using:
\[ A = \sqrt{s(s - a)(s - b)(s - c)} \]
where \(a = AB = 48\), \(b = BC = 14\), and \(c = AC = 50\).
\[ A = \sqrt{56(56 - 48)(56 - 14)(56 - 50)} \]
\[ A = \sqrt{56 \times 8 \times 42 \times 6} \]

4. **Calculate the area**:
- First, calculate the products:
\[ 56 \times 8 = 448 \]
\[ 42 \times 6 = 252 \]
\[ A = \sqrt{448 \times 252} \]

5. **Finding the square root**:
- Simplifying further:
\[ A = \sqrt{112896} = 336 \text{ cm}^2 \]

6. **Total area of parallelogram ABCD**:
- Since the area of triangle ABC is equal to the area of triangle ADC, the total area of the parallelogram is:
\[ \text{Area of ABCD} = 2 \times 336 = 672 \text{ cm}^2 \]

### Step 2: Find the distance between the longer sides (height from BC to AD)

1. **Using the area formula for parallelogram**:
\[ \text{Area} = \text{base} \times \text{height} \]
- Here, base = AB = 48 cm, and we need to find height \(h_1\):
\[ 672 = 48 \times h_1 \]
\[ h_1 = \frac{672}{48} = 14 \text{ cm} \]

### Step 3: Find the distance between the shorter sides (height from AD to BC)

1. **Using the area formula again**:
- Here, base = BC = 14 cm, and we need to find height \(h_2\):
\[ 672 = 14 \times h_2 \]
\[ h_2 = \frac{672}{14} = 48 \text{ cm} \]

### Final Answers:
(i) The area of the parallelogram ABCD is **672 cm²**.
(ii) The distance between the longer sides (height from BC to AD) is **14 cm**.
(iii) The distance between the shorter sides (height from AD to BC) is **48 cm**.