ABCD is a trapezium in which AB//DC and AB=70cm DC=50cm. If P and Q are respectively mid points of AD and BC. Find PQ and show that ar(trapezium DCQP)=9/11ar(trapezium ABQP)
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Find d(A,B), if coordinate of A and B are -2 and 5 respectively. *
To find the length of PQ and prove the given ratio of areas, we can follow these steps:
1. Draw the trapezium ABCD and mark the points P and Q as the midpoints of AD and BC, respectively.
2. Since PQ is a line segment connecting the midpoints of the sides, we know that PQ is parallel to AB and DC, and its length is half the length of the sum of AB and DC.
- PQ = 1/2 * (AB + DC)
- Substituting the given values, PQ = 1/2 * (70 + 50) = 1/2 * 120 = 60 cm.
Now, let's prove that the ratio of the areas of trapezium DCQP and trapezium ABQP is 9/11.
3. Start by calculating the area of trapezium ABQP.
- The area of a trapezium is given by the formula: Area = 1/2 * (sum of parallel sides) * height.
- In trapezium ABQP, the sum of parallel sides is AB + PQ.
- The height of the trapezium is the perpendicular distance between AB and PQ. Since AB is parallel to DC, the height will be the same as the perpendicular distance between DC and PQ.
- Therefore, the area of trapezium ABQP = 1/2 * (AB + PQ) * (perpendicular distance between DC and PQ).
4. Now, calculate the area of trapezium DCQP.
- In trapezium DCQP, the sum of the parallel sides is DC + PQ.
- The height of the trapezium is the perpendicular distance between DC and PQ, which is the same as the height of trapezium ABQP.
- Therefore, the area of trapezium DCQP = 1/2 * (DC + PQ) * (perpendicular distance between DC and PQ).
5. To prove that the ratio of the areas is 9/11, we need to compare the expressions for the areas of both trapeziums.
- Area of trapezium DCQP/Area of trapezium ABQP = (1/2 * (DC + PQ) * (perpendicular distance between DC and PQ)) / (1/2 * (AB + PQ) * (perpendicular distance between DC and PQ)).
- Simplifying the expression:
- Area of trapezium DCQP/Area of trapezium ABQP = (DC + PQ) / (AB + PQ).
- Substituting the values: (DC + PQ) / (AB + PQ) = (50 + 60) / (70 + 60) = 110 / 130 = 11/13.
Since the ratio (11/13) obtained in step 5 is not equal to the given ratio (9/11), we can conclude that there might be an error in the given question or a mistake in the calculations.