In parallelogram ABCD, P is a point on side AB and Q is a point on side BC. Prove that : (i) △ CPD and △ AQD are equal in area. (ii) Area (△ AQD) = Area (△ APD) + Area (△ CPB)

We know that, Area of a triangle is half that of a parallelogram on the same base and between the same parallels. (i) From figure, △ CPD and || gm ABCD are on the same base CD and between the same parallel lines AB and CD. ∴ Area of triangle CPD = Area of parallelogram ABCD ........(1) △ AQD and || gm ABCD are on the same base AD and between the same parallel lines AD and BC. ∴ Area of triangle AQD = Area of parallelogram ABCD ........(2) From equations (1) and (2), we get : Area of triangle CPD = Area of triangle AQD. Hence, proved that area of triangle CPD = area of triangle AQD. (ii) From part (i), we get : ⇒ Area of △ CPD = Area of parallelogram ABCD ∴ Area of || gm ABCD - Area of △ CPD = Area of parallelogram ABCD ........(3) From figure, ⇒ Area of || gm ABCD - Area of △ CPD = Area of △ APD + Area of △ CPB ............(4) From equations (3) and (4), we get : ⇒ Area of △ APD + Area of △ CPB = Area of || gm ABCD ........(5) Since, ⇒ Area of △ ADQ = Area of || gm ABCD [From eqn. 2] Substituting value of Area of || gm ABCD from above equation in equation (5), we get : ⇒ Area of △ APD + Area of △ CPB = Area of △ ADQ. Hence, proved that area (△ AQD) = area (△ APD) + area (△ CPB).

The given figure shows a rectangle ABDC and a parallelogram ABEF; drawn on opposite sides of AB. Prove that :
(i) quadrilateral CDEF is a parallelogram
(ii) Area of quad. CDEF = Area of rect. ABDC + Area of // gm ABEF

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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.

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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.

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Mathematics

In parallelogram ABCD, P is a point on side AB and Q is a point on side BC. Prove that :
(i) △ CPD and △ AQD are equal in area.
(ii) Area (△ AQD) = Area (△ APD) + Area (△ CPB)

Theorems on Area

Related Questions

The given figure shows a rectangle ABDC and a parallelogram ABEF; drawn on opposite sides of AB. Prove that :
(i) quadrilateral CDEF is a parallelogram
(ii) Area of quad. CDEF = Area of rect. ABDC + Area of // gm ABEF

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.

Mathematics

In parallelogram ABCD, P is a point on side AB and Q is a point on side BC. Prove that :(i) △ CPD and △ AQD are equal in area.(ii) Area (△ AQD) = Area (△ APD) + Area (△ CPB)

Theorems on Area

Related Questions

The given figure shows a rectangle ABDC and a parallelogram ABEF; drawn on opposite sides of AB. Prove that :(i) quadrilateral CDEF is a parallelogram(ii) Area of quad. CDEF = Area of rect. ABDC + Area of // gm ABEF

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.

In parallelogram ABCD, P is a point on side AB and Q is a point on side BC. Prove that :
(i) △ CPD and △ AQD are equal in area.
(ii) Area (△ AQD) = Area (△ APD) + Area (△ CPB)

The given figure shows a rectangle ABDC and a parallelogram ABEF; drawn on opposite sides of AB. Prove that :
(i) quadrilateral CDEF is a parallelogram
(ii) Area of quad. CDEF = Area of rect. ABDC + Area of // gm ABEF

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.