# Class 9 Areas of Parallelograms and Triangles Important Questions

Important Questions**Areas of Parallelograms and Triangles Important Questions**

**If AD is one of the medians of a triangle and P is a point on AD. Prove that****ar(∆BDP) = ar(∆CDP)****ar(∆ABP) = ar(∆ACP)**

**In the adjoining figure, D is the mid-point of side AB of ∆ABC and P is any point on BC. If CQ||PD meets AB in Q, prove that ar(∆BPQ) = 1/2 ar(∆ABC).****A point O inside a rectangle ABCD is joined to the vertices. Prove that the sum of the areas of a pair of opposite triangles so formed is equal to the sum of the areas of the other pair of triangles.****D, E, F are the midpoints of the sides BC, CA and AB respectively of****∆ ABC. Prove that****BDEF is a parallelogram,****ar(∆DEF) = 1/4 ar(∆ABC) and****ar(****||**gm**BDEF) =1/2ar(∆ABC).**

**In the adjoining figure, PQRS and PABC are two parallelograms of equal area. Prove that QC||BR.****X and Y are points on the side LN of the triangle LMN, such that LX = XY = YN. Through X, a line is drawn parallel to LM to meet MN at Z(see figure). Prove that ar(****∆**LZY) = ar (quad. MZYX).**ABCD is a parallelogram in which BC is produced to E such the CE = BC. AE intersects CD at F (see figure). IF area of****∆**DFB = 3 cm^{2}, find the area of the parallelogram ABCD.**ABCD is a parallelogram and P is any point in its interior. Show that ar(****∆****APB) + ar (**CPD) = ar(**∆**BPC) + ar(**∆**APD).**∆****ABCD is a parallelogram and BC is produced to a point Q such that AD= CQ. If AQ intersects DC at P, show that area of ΔBPC= area of ΔDPQ.****ABCD is a trapezium in which AB is parallel to DC, DC = 40 cm and AB = 60 cm. If X and Y are the mid – points of AD and BC****respectively**, prove that:**XY = 50 cm****DCYX is a trapezium****ar (trap. DCYX) = (9/11)ar (trap.(XYBA).**

**Let P ,Q ,R ,S be respectively the mid points of the sides AB, BC, CD and DA of quadrilateral ABCD . Show that PQRS is a parallelogram such that ar(parallelogram PQRS) = 1/2 ar(quad. ABCD).****The area of parallelogram PQRS is 88 c****m**. A perpendicular from S is drawn to intersect PQ at M. If SM = 8 cm, then find the length of PQ.^{2}**ABCD is a parallelogram and O is a point in its interior. Prove that****ar (∆AOB) + ar(****∆**COD) = 1/2 ar (parallelogram ABCD).**In Figure, diagonals AC and BD of quadrilateral ABCD intersect at O such that OB = OD.****If AB = CD, then show that:****ar (****∆**DOC) = ar (**∆**AOB)**ar (****∆**DCB) = ar (**∆**ACB)**DA || CB or ABCD is a parallelogram.**

**In the adjoining figure, ABCD is a parallelogram and a line through A cuts DC at P and BC produced at Q. Prove that ar(****∆**BPC) = ar(**∆**DPQ).

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