Find the area of a quadrilateral with sides 17,25, 30 and 28 cm and one of its diagonal is 26 cm:

Options

A. `450cm^(2)`
B. `360cm^(2)`
C. `540cm^(2)`
D. `720cm^(2)`

Answer

To find the area of the quadrilateral with sides 17 cm, 25 cm, 30 cm, and 28 cm, and one of its diagonals measuring 26 cm, we can divide the quadrilateral into two triangles using the diagonal. We will then calculate the area of each triangle separately and sum them up.

### Step-by-Step Solution:

1. **Label the Quadrilateral**:
Let's label the quadrilateral as ABCD, where:
- AB = 17 cm
- BC = 25 cm
- CD = 30 cm
- DA = 28 cm
- Diagonal AC = 26 cm

2. **Calculate the Area of Triangle ABC**:
- The sides of triangle ABC are AB = 17 cm, AC = 26 cm, and BC = 25 cm.
- First, calculate the semi-perimeter (s) of triangle ABC:
\[ s = \frac{AB + AC + BC}{2} = \frac{17 + 26 + 25}{2} = \frac{68}{2} = 34 \text{ cm} \]
- Now, use Heron's formula to find the area (A) of triangle ABC:
\[ A = \sqrt{s(s - AB)(s - AC)(s - BC)} \]
\[ A = \sqrt{34(34 - 17)(34 - 26)(34 - 25)} \]
\[ A = \sqrt{34 \times 17 \times 8 \times 9} \]
- Calculate the area:
\[ A = \sqrt{34 \times 17 \times 72} = \sqrt{34 \times 1224} = \sqrt{41616} \]
\[ A \approx 204 \text{ cm}^2 \]

3. **Calculate the Area of Triangle ACD**:
- The sides of triangle ACD are AC = 26 cm, AD = 28 cm, and CD = 30 cm.
- Calculate the semi-perimeter (s) of triangle ACD:
\[ s = \frac{AC + AD + CD}{2} = \frac{26 + 28 + 30}{2} = \frac{84}{2} = 42 \text{ cm} \]
- Use Heron's formula to find the area (A) of triangle ACD:
\[ A = \sqrt{s(s - AC)(s - AD)(s - CD)} \]
\[ A = \sqrt{42(42 - 26)(42 - 28)(42 - 30)} \]
\[ A = \sqrt{42 \times 16 \times 14 \times 12} \]
- Calculate the area:
\[ A = \sqrt{42 \times 2688} = \sqrt{112896} \]
\[ A \approx 336 \text{ cm}^2 \]

4. **Total Area of Quadrilateral ABCD**:
- Now, add the areas of triangles ABC and ACD:
\[ \text{Total Area} = \text{Area of } ABC + \text{Area of } ACD \]
\[ \text{Total Area} = 204 + 336 = 540 \text{ cm}^2 \]

### Final Answer:
The area of the quadrilateral ABCD is **540 cm²**.