In the given figure, diagonals PR and QS of the parallelogram PQRS intersect at point O and LM is parallel to PS. Show that : (i) 2 Area (△ POS) = Area (// gm PMLS) (ii) Area (△ POS) + Area (△ QOR) = 1/2 Area (//gm PQRS) (iii) Area (△ POS) + Area (△ QOR) = Area (△ POQ) + Area (△ SOR)

Question

KnowledgeBoat · Accepted Answer

(i) We know that, Area of a triangle is half that of a parallelogram on the same base and between the same parallels. From figure, Triangle POS and parallelogram PMLS lie on same base PS and between same parallel lines PS and ML. ∴ Area of triangle POS = Area of parallelogram PMLS ⇒ Area of parallelogram PMLS = 2 (Area of triangle POS) Hence, proved that 2 area (△ POS) = area (// gm PMLS). (ii) We know that, Area of a triangle is half that of a parallelogram on the same base and between the same parallels. From figure, Triangle POS and parallelogram PMLS lie on same base PS and between same parallel lines PS and ML. ∴ Area of triangle POS = Area of parallelogram PMLS ..........(1) Triangle QOR and parallelogram MQRL lie on same base QR and between same parallel lines QR and ML. ∴ Area of triangle QOR = Area of parallelogram MQRL ..........(2) Adding equations (1) and (2), we get : ⇒ Area of triangle POS + Area of triangle QOR = Area of parallelogram PMLS + Area of parallelogram MQRL ⇒ Area of triangle POS + Area of triangle QOR = (Area of parallelogram PMLS + Area of parallelogram MQRL) ⇒ Area of triangle POS + Area of triangle QOR = Area of parallelogram PQRS. Hence, proved that area (△ POS) + area (△ QOR) = area (//gm PQRS). (iii) We know that, In a parallelogram diagonals bisect each other. ∴ OS = OQ In △ PQS, O is the mid-point of QS. ∴ OP is the median. We know that, Median of a triangle divides it into two triangles of equal areas. ∴ Area of △ POS = Area of △ POQ .........(3) In △ QSR, O is the mid-point of QS. ∴ OR is the median. We know that, Median of a triangle divides it into two triangles of equal areas. ∴ Area of △ QOR = Area of △ SOR .........(4) Adding equations (3) and (4), we get : ⇒ Area of △ POS + Area of △ QOR = Area of △ POQ + Area of △ SOR. Hence, proved that area (△ POS) + area (△ QOR) = area (△ POQ) + area (△ SOR).

Mathematics

In the given figure, diagonals PR and QS of the parallelogram PQRS intersect at point O and LM is parallel to PS. Show that : (i) 2 Area (△ POS) = Area (// gm PMLS) (ii) Area (△ POS) + Area (△ QOR) = 1/2 Area (//gm PQRS) (iii) Area (△ POS) + Area (△ QOR) = Area (△ POQ) + Area (△ SOR)

Theorems on Area

Related Questions

In the given figure, if area of triangle ADE is 60 cm²; state, giving reason, the area of :
(i) parallelogram ABED;
(ii) rectangle ABCF;
(iii) triangle ABE.

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 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)

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.

Mathematics

In the given figure, diagonals PR and QS of the parallelogram PQRS intersect at point O and LM is parallel to PS. Show that : (i) 2 Area (△ POS) = Area (// gm PMLS) (ii) Area (△ POS) + Area (△ QOR) = 1/2 Area (//gm PQRS) (iii) Area (△ POS) + Area (△ QOR) = Area (△ POQ) + Area (△ SOR)

Theorems on Area

Related Questions

In the given figure, if area of triangle ADE is 60 cm2; state, giving reason, the area of :(i) parallelogram ABED;(ii) rectangle ABCF;(iii) triangle ABE.

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 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)

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 given figure, if area of triangle ADE is 60 cm²; state, giving reason, the area of :
(i) parallelogram ABED;
(ii) rectangle ABCF;
(iii) triangle ABE.

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 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)

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.