a In the figure 1 given below ABCD is a parallelogram and P is any point in BC Prove that Area of AB...

Question

a In the figure 1 given below ABCD is a parallelogram and P is any point in BC Prove that Area of ABP area of DPC Area of APD b In the figure 2 given below O is any point inside a parallelogram ABCD Prove that i area of OAB area of OCD area of gm ABCD ii area of OBC area of OAD area of gm ABCD

Accepted Answer

a Given From fig 1 ABCD is a parallelogram and P is any point in BC To prove Area of ABP area of DPC Area of APD Proof APD and gm ABCD are on the same base AD and between the same lines AD and BC ar APD ar gm ABCD 1 In parallelogram ABCD ar gm ABCD ar ABP ar APD ar DPC Now divide both sides by 2 we get ar gm ABCD ar ABP ar APD ar DPC 2 From 1 and 2 ar APD ar gm ABCD Substituting 2 in 1 ar APD ar ABP ar APD ar DPC ar APD ar APD ar ABP ar DPC ar APD ar ABP ar DPC ar APD ar ABP ar DPC Or ar ABP ar DPC ar APD Hence proved b Given From fig 2 gm ABCD in which O is any point inside it To prove i area of OAB area of OCD area of gm ABCD ii area of OBC area of OAD area of gm ABCD Draw POQ AB through O It meets AD at P and BC at Q Proof i AB PQ and AP BQ ABQP is a gm Similarly PQCD is a gm Now OAB and gm ABQP are on same base AB and between same lines AB and PQ ar OAB ar gm ABQP 1 Similarly ar OCD ar gm PQCD 2 Now by adding 1 and 2 ar OAB ar OCD ar gm ABQP ar gm PQCD ar gm ABQP ar gm PQCD ar gm ABCD ar OAB ar OCD ar gm ABCD Hence proved ii we know that ar OAB ar OBC ar OCD ar OAD ar gm ABCD ar OAB ar OCD ar OBC ar OAD ar gm ABCD ar gm ABCD ar OBC ar OAD ar gm ABCD ar OBC ar OAD ar gm ABCD ar gm ABCD ar OBC ar OAD ar gm ABCD Hence proved

ML Aggarwal Solutions Class 9 Mathematics Solutions for Theorems on Area Exercise 14 in Chapter 14 - Theorems on Area