Q20 of 77 Page 15

In Fig. 15.83, CD||AE and CY||BA.

(i) Name a triangle equal in area of Δ CBX



(ii) Prove that ar(Δ ZDE) = ar(Δ CZA)


(iii) Prove that ar(Δ BCYZ) = ar(Δ EDZ)

(i) ΔAYC and Δ BCY are on the same base CY and between the same parallels

CY || AB


Area (ΔAYC) = Area (ΔBCY)


(Triangles on the same base and between the same parallels are equal in area)


Subtracting ΔCXY from both sides we get,


Area (ΔAYC) – Area (ΔCXY) = Area (ΔBCY) – Area (ΔCXY) (Equals subtracted from equals are equals)


Area (ΔCBX) = Area (ΔAXY)


(ii) Since, ΔACC and ΔADE are on the same base AF and between the same parallels


CD || AF


Then,


Area ( = Area ()


Area () + Area ( = Area () + Area (


Area ( = Area () (i)


(iii) Since, ΔCBY and ΔCAY are on the same base CY and between the same parallels


CY || BA


Then,


Area () = Area ()


Adding Area ( on both sides we get


Area ( + Area ( = Area ( + Area ()


Area ( = Area ( (ii)


Compare (i) and (ii), we get


Area ( = Area (


More from this chapter

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18

In a Δ ABC, P and Q are respectively the mid-point of AB and BC and R is the mid-point of AP. Prove that:

(i) ar(Δ PBQ) = ar(Δ ARC)


(ii) ar(Δ PQR) = ar(Δ ARC)


(iii) ar(Δ RQC) = ar(Δ ABC)

 19

ABCD is a parallelogram, G is the point on AB such that AG = 2GB, E is a point of DC such that CE = 2DE and F is the point of BC such that BF=2FC. Prove that:

(i) ar(Δ ADGE) = ar(Δ GBCE)


(ii) ar(Δ EGB) = ar(Δ ABCD)


(iii) ar(Δ EFC) = ar(Δ EBF)


(iv) ar(Δ EBG) = ar(Δ EFC)

 21

In Fig. 15.84, PSDA is a parallelogram in which PQ=QR=RS and AP||BQ||CR. Prove that


ar(Δ PQE) = ar(Δ CFD)

 22

In Fig. 15.85, ABCD is a trapezium in which AB||DC and DC=40 cm and AB=60 cm. If X and Y are, respectively, the mid points of AD and BC, Prove that


(i) XY =50 cm (ii) DCYX is a trapezium


(iii) ar(trap. DCYX) = ar(trap. XYBA)