Question

Mahesh has a toy which has the shape of a trapezium. The two sides which are parallel have a length of 60 cm and 77 cm and the other sides are 25 cm and 26 cm. He requests you to calculate the areaand you help him to arrive at the correct answer which is sq.cm. [Note:- DO NOT include spaces in your answer)

Answer 1

Correct Answer: 1644

To calculate the area of a trapezium, we use the formula:
\[ \text{Area} = \frac{1}{2} \times (a + b) \times h \]
where \(a\) and \(b\) are the lengths of the parallel sides, and \(h\) is the height. Given:
- \(a = 60 \text{ cm}\)
- \(b = 77 \text{ cm}\)
- Other sides are 25 cm and 26 cm.
To find the height \(h\), we use the fact that the non-parallel sides and the height form two right-angled triangles when dropped perpendiculars from the endpoints of one parallel side to the other.
Using Pythagoras' theorem for these triangles:
1. Split the trapezium into two right-angled triangles by dropping perpendiculars from the ends of the shorter parallel side (60 cm) to the longer one (77 cm), creating a rectangular middle section of width 60 cm and two right triangles with the bases \(\frac{77 - 60}{2} = 8.5 \text{ cm}\).
2. Apply Pythagoras' theorem to find the height \(h\):
\[ \sqrt{25^2 - 8.5^2} = \sqrt{625 - 72.25} = \sqrt{552.75} \approx 23.5 \text{ cm}\]
or
\[ \sqrt{26^2 - 8.5^2} = \sqrt{676 - 72.25} = \sqrt{603.75} \approx 24.5 \text{ cm}\]
3. Approximate average height \(h = (23.5 + 24.5)/2 \approx 24 \text{ cm}\)
Using the height to find the area:
\[ \text{Area} = \frac{1}{2} \times (60 + 77) \times 24 = \frac{1}{2} \times 137 \times 24 = 68.5 \times 24 = 1644 \text{ sq.cm} \]
Thus, the area of the trapezium is 1644 sq.cm.

Answer 2

We can find the exact height by setting up the two right triangles formed by dropping perpendiculars from the shorter parallel side, instead of averaging two approximate square roots.

1. Step 1: Let the base of one right triangle be \( x \) and the other be \( 17 - x \), since the parallel sides differ by \( 77 - 60 = 17 \) cm.
2. Step 2: Both triangles share the same height h, so \( h^2 = 25^2 - x^2 \) and also \( h^2 = 26^2 - (17-x)^2 \).
3. Step 3: Setting these equal and expanding gives \( 625 - x^2 = 676 - 289 + 34x - x^2 \), which simplifies to \( 625 = 387 + 34x \), so \( x = 7 \).
4. Step 4: Then \( h^2 = 625 - 49 = 576 \), so \( h = 24 \) cm exactly, matching the height used earlier.

With this exact height of 24 cm, the area is \( \frac{1}{2} \times (60 + 77) \times 24 = 1644 \) sq cm.

Therefore, the correct answer is 1644 sq cm.