If area of An Trapezium (is oceles) is 120 `cm^(2)` height = 8 cm find its diagonal

Answer

To find the diagonal of an isosceles trapezium given its area and height, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Properties of the Isosceles Trapezium**:
- An isosceles trapezium has two parallel sides (let's call them AB and CD) and two equal non-parallel sides (AD and BC).
- The height (h) is the perpendicular distance between the two parallel sides.

2. **Use the Area Formula**:
- The area (A) of a trapezium can be calculated using the formula:
\[ A = \frac{1}{2} \times (a + b) \times h \]
where \(a\) and \(b\) are the lengths of the parallel sides, and \(h\) is the height.

3. **Substitute Known Values**:
- We know the area \(A = 120 \, cm^2\) and the height \(h = 8 \, cm\).
- Plugging these values into the area formula gives:
\[ 120 = \frac{1}{2} \times (a + b) \times 8 \]

4. **Solve for the Sum of the Parallel Sides**:
- Rearranging the equation:
\[ 120 = 4 \times (a + b) \]
\[ a + b = \frac{120}{4} = 30 \]

5. **Visualize the Trapezium**:
- Since it is isosceles, we can drop perpendiculars from points B and C to line AD, creating two right triangles on either side of the trapezium.

6. **Determine the Length of the Base**:
- The length of the base (the distance between the feet of the perpendiculars) will be \(a\) and \(b\) where \(a + b = 30\).
- For simplicity, let’s assume \(a = b\) (which is a common case for isosceles trapeziums). Thus, \(a = b = 15 \, cm\).

7. **Calculate the Diagonal**:
- Now, we can form a right triangle using the height and half the difference of the bases:
- The height \(h = 8 \, cm\) and half the base difference is \( \frac{15 - 15}{2} = 0 \, cm\).
- The diagonal (d) can be calculated using the Pythagorean theorem:
\[ d = \sqrt{h^2 + \left(\frac{a - b}{2}\right)^2} \]
Since \(a = b\), we have:
\[ d = \sqrt{8^2 + 15^2} = \sqrt{64 + 225} = \sqrt{289} = 17 \, cm \]

### Final Answer:
The diagonal of the isosceles trapezium is **17 cm**.