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Areas of Parallelograms and Triangles

Long Answer Type

41. 8. XY is a line parallel to side BC of a triangle ABC. If BE || AC and CF || AB meet XY at E and F respectively, show that ar(ΔABE) = ar(ΔACF).

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42. The side AB of a parallelogram ABCD is produced to any point P. A line through A and parallel to CP meets CB produced at Q and then parallelogram PBQR is completed. Show that ar(||gm ABCD) = ar(||gm PBQR).

[Hint. Join AC and PQ. Now compare ar(ACQ) and ar(APQ)].

Given: The side AB of a parallelogram ABCD is produced to any point P. A line through A and parallel to CP meets CB produced at Q and then parallelogram PBQR is completed.

To Prove: ar(|| gm ABCD) = ar(|| gm PBQR). Construction: Join AC and PQ.
Proof: ∵ AC is a diagonal of || gm ABCD

$therefore space ar left parenthesis increment ABC right parenthesis equals 1 half ar left parenthesis parallel to space gm space ABCD right parenthesis space space space space space space space space space space space space space space... left parenthesis 1 right parenthesis$
$because$ PQ is a diagonal of $parallel to$ gm BQRP
$therefore space space ar left parenthesis increment BPQ right parenthesis equals 1 half ar left parenthesis space parallel to space gm space BQRp right parenthesis space space space space space space space space space space space space space... left parenthesis 2 right parenthesis$

∵ ΔACQ and ΔAPQ are on the same base AQ and between the same parallels AQ and CP. ar(ΔACQ) = ar(ΔAPQ)
∵ Two triangles on the same base (or equal bases) and between the same parallels are equal in area
⇒ ar(ΔACQ) – ar(ΔABQ)
= ar(ΔAPQ) – ar(ΔABQ)
| Subtracting the same areas from both sides ⇒ ar(ΔABC) = ar(ΔBPQ)

$rightwards double arrow space 1 half ar left parenthesis space parallel to space gm space ABCD right parenthesis space equals space 1 half space ar left parenthesis space parallel to space gm space PBQR right parenthesis$

| From (1) and (2)
⇒ ar(|| gm ABCD) = ar(|| gm PBQR).

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Short Answer Type

43. Diagonals AC and BD of a trapezium ABCD with AB || DC intersect each other at O. Prove that ar(ΔAOD) = ar(ΔBOC).

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44.

In figure, ABCDE is a pentagon. A line through B parallel to AC meets DC produced at F. Show that
(i) ar(ΔACB) = ar(ΔACF)
(ii) ar(□AEDF) = ar(ABCDE).

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45. A villager Itwaari has a plot of land of the shape of a quadrilateral. The Gram Panchayat of the village decided to take over some portion of his plot from one of the comers to construct a Health Centre. Itwaari agrees to the above proposal with the condition that he should be given equal amount of land in lieu of his land adjoining his plot so as to form a triangular plot. Explain how this proposal will be implemented.

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46. ABCD is a trapezium with AB || DC. A line parallel to AC intersects AB at X and BC at Y. Prove that ar(ΔADX) = ar(ΔACY).

[Hint. Join CX.]

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47. In figure, AP || BQ || CR. Prove that ar(ΔAQC) = ar(ΔPBR).

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48. Diagonals AC and BD of a quadrilateral ABCD intersect at O in such a way that ar(ΔAOD) = ar(ΔBOC). Prove that ABCD is a trapezium.

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49. Infigure, ar(ΔDRC) = ar(ΔDPC) and ar(ΔBDP) = ar(ΔARC). Show that both the quadrilateral ABCD and DCPR are trapeziums.

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50.

Show that the area of a rhombus is half the product of the lengths of its diagonals.

Prove that the area of a rhombus is equal to half the rectangle contained by its diagonals.

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