ABCD is a parallelogram, a line through A cuts DC at point P and BC produced at Q. Prove that triangle BCP is equal in area to triangle DPQ.
Theorems on Area
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Question

ABCD is a parallelogram, a line through A cuts DC at point P and BC produced at Q. Prove that triangle BCP is equal in area to triangle DPQ.

KnowledgeBoat · Accepted Answer

We know that, Area of a triangle is half that of a parallelogram on the same base and between the same parallels. Since, triangle APB and parallelogram ABCD are on the same base AB and between the same parallels AB and DC. ∴ Area of △ APB = Area of || gm ABCD ........(1) Since, triangle ADQ and parallelogram ABCD are on the same base AD and between the same parallels AD and BQ. ∴ Area of △ ADQ = Area of || gm ABCD ........(2) Adding equations (1) and (2), we get : ⇒ Area of △ APB + Area of △ ADQ = Area of || gm ABCD + Area of || gm ABCD ⇒ Area of △ APB + Area of △ ADQ = Area of || gm ABCD ...........(3) From figure, ⇒ Area of △ APB + Area of △ ADQ = Area of quadrilateral ADQB - Area of △ BPQ ..............(4) From equation (3) and (4), we get : ⇒ Area of quadrilateral ADQB - Area of △ BPQ = Area of || gm ABCD ⇒ Area of quadrilateral ADQB - Area of △ BPQ = Area of quadrilateral ADQB - Area of △ DCQ ⇒ Area of △ BPQ = Area of △ DCQ ⇒ Area of △ BPQ - Area of △ PCQ = Area of △ DCQ - Area of △ PCQ ⇒ Area of △ BCP = Area of △ DPQ. Hence, proved that area of △ BCP = area of △ DPQ.

Mathematics

ABCD is a parallelogram, a line through A cuts DC at point P and BC produced at Q. Prove that triangle BCP is equal in area to triangle DPQ.

Theorems on Area

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Mathematics

ABCD is a parallelogram, a line through A cuts DC at point P and BC produced at Q. Prove that triangle BCP is equal in area to triangle DPQ.

Theorems on Area

Related Questions

In the given figure, M and N are the mid-points of the sides DC and AB respectively of the parallelogram ABCD.If the area of parallelogram ABCD is 48 cm2;(i) state the area of the triangle BEC.(ii) name the parallelogram which is equal in area to the triangle BEC.

In the following figure, CE is drawn parallel to diagonal DB of the quadrilateral ABCD which meets AB produced at point E.Prove that △ ADE and quadrilateral ABCD are equal in area.

The given figure shows a pentagon ABCDE. EG drawn parallel to DA meets BA produced at G and CF drawn parallel to DB meets AB produced at F. Prove that the area of pentagon ABCDE is equal to the area of triangle GDF.

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In the given figure, M and N are the mid-points of the sides DC and AB respectively of the parallelogram ABCD.
If the area of parallelogram ABCD is 48 cm²;
(i) state the area of the triangle BEC.
(ii) name the parallelogram which is equal in area to the triangle BEC.

In the following figure, CE is drawn parallel to diagonal DB of the quadrilateral ABCD which meets AB produced at point E.
Prove that △ ADE and quadrilateral ABCD are equal in area.