In parallelogram ABCD, P is mid-point of AB. CP and BD intersect each other at point O. If area of △ POB = 40 cm 2 and OP : OC = 1 : 2, find : (i) Areas of △ BOC and △ PBC (ii) Area of △ ABC and parallelogram ABCD.

(i) Given, OP : OC = 1 : 2 We know that, Ratio of the area of triangles with same vertex and bases along the same line is equal to the ratio of their respective bases. From figure, Area of △ PBC = Area of △ BOC + Area of △ POB = 80 + 40 = 120 cm 2 . Hence, area of △ BOC = 80 cm 2 and area of △ PBC = 120 cm 2 . (ii) Given, P is the mid-point of AB. ∴ CP is the median of △ ABC. We know that, Median of triangle divides it into two triangles of equal area. ∴ Area of △ APC = Area of △ PBC = 120 cm 2 . From figure, ⇒ Area of △ ABC = Area of △ APC + Area of △ BPC = 120 + 120 = 240 cm 2 . We know that, The area of triangle is half that of a parallelogram on the same base and between the same parallels. Since, △ ABC and || gm ABCD lies on same base AB and between same parallel lines AB and DC. ∴ Area of △ ABC = Area of || gm ABCD ⇒ Area of || gm ABCD = 2 Area of △ ABC = 2 × 240 = 480 cm 2 . Hence, area of △ ABC = 240 cm 2 and || gm ABCD = 480 cm 2 .

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Mathematics

In parallelogram ABCD, P is mid-point of AB. CP and BD intersect each other at point O. If area of △ POB = 40 cm² and OP : OC = 1 : 2, find :
(i) Areas of △ BOC and △ PBC
(ii) Area of △ ABC and parallelogram ABCD.

Theorems on Area

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Mathematics

In parallelogram ABCD, P is mid-point of AB. CP and BD intersect each other at point O. If area of △ POB = 40 cm² and OP : OC = 1 : 2, find :
(i) Areas of △ BOC and △ PBC
(ii) Area of △ ABC and parallelogram ABCD.

Theorems on Area

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Mathematics

In parallelogram ABCD, P is mid-point of AB. CP and BD intersect each other at point O. If area of △ POB = 40 cm2 and OP : OC = 1 : 2, find :(i) Areas of △ BOC and △ PBC(ii) Area of △ ABC and parallelogram ABCD.

Theorems on Area

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The perimeter of a triangle ABC is 37 cm and the ratio between the lengths of its altitudes be 6 : 5 : 4. Find the lengths of its sides.

In parallelogram ABCD, P is mid-point of AB. CP and BD intersect each other at point O. If area of △ POB = 40 cm² and OP : OC = 1 : 2, find :
(i) Areas of △ BOC and △ PBC
(ii) Area of △ ABC and parallelogram ABCD.