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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)].

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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.

Or

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

Let ABCD be a rhombus whose diagonals are AC and BD.
Then,
Area of rhombus ABCD
= Area of ΔABD + Area of ΔCBD

$equals space fraction numerator left parenthesis BD right parenthesis left parenthesis AO right parenthesis over denominator 2 end fraction plus fraction numerator left parenthesis BD right parenthesis left parenthesis OC right parenthesis over denominator 2 end fraction$
∵ Diagonals of a rhombus are perpendiculars to each other

Let ABCD be a rhombus whose diagonals are AC and BD.Then,Area of rhom

$equals fraction numerator left parenthesis BD right parenthesis over denominator 2 end fraction left parenthesis AO plus OC right parenthesis equals fraction numerator left parenthesis BD right parenthesis left parenthesis AC right parenthesis over denominator 2 end fraction equals space 1 half$

Product of the lengths of its diagonals.