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Areas of Parallelograms and Triangles

Short Answer Type

1. Which of the following figures lie on the same base and between the same parallels. In such a case, write the common base and the two parallels.

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2. In figure, ABCD is a parallelogram, AE ⊥ DC and CF ⊥ AD. If AB = 16 cm, AE = 8 cm and CF = 10 cm, find AD.

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Long Answer Type

If E, F, G and H are respectively the mid-points of the sides of a parallelogram ABCD, show that ar(EFGH) = $1 half$ ar(ABCD).

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Short Answer Type

4. P and Q are any two points lying on the sides DC and AD respectively of a parallelogram ABCD. Show that ar(ΔAPB) = ar{ΔBQC).

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5. In figure, P is a point in the interior of a parallelogram ABCD. Show that:

(1) ar(

ar left parenthesis increment APB right parenthesis plus ar left parenthesis increment PCD right parenthesis equals 1 half ar left parenthesis space parallel to space gm space ABCD right parenthesis

(ii) ar(ΔAPD) + ar(ΔPBC) = ar(ΔAPB) + ar(ΔPCD). [CBSE 2012 (March)]

[Hint. Through P, draw a line parallel to AB.]

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Long Answer Type

In figure, PQRS and ABRS are parallelograms and X is any point on side BR. Show that:

(i) ar(|| gm PQRS) = ar(|| gm ABRS)

(ii) $ar left parenthesis increment AXS right parenthesis equals 1 half ar left parenthesis space parallel to space gm space PQRS right parenthesis.$

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Short Answer Type

7. A farmer was having a field in the form of a parallelogram PQRS. She took any point A on RS and joined it to points P and Q. In how many parts the fields is divided? What are the shapes of these parts? The farmer wants to sow wheat and pulses in equal portions of the field separately. How should she do it?

∵ ΔAPQ, ΔAPS and ΔAQR lie between the same parallels

∴ Their altitudes are same. Let it be x. Then,

a r left parenthesis increment A P Q right parenthesis equals fraction numerator left parenthesis P Q right parenthesis left parenthesis x right parenthesis over denominator 2 end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 1 right parenthesis a r left parenthesis increment A P S right parenthesis plus a r left parenthesis increment A Q R right parenthesis space space space space space space space space space space space equals space fraction numerator left parenthesis A S right parenthesis left parenthesis x right parenthesis over denominator 2 end fraction plus fraction numerator left parenthesis A R right parenthesis left parenthesis x right parenthesis over denominator 2 end fraction space space space space space space space space space space space equals space fraction numerator left parenthesis A S plus A R right parenthesis left parenthesis x right parenthesis over denominator 2 end fraction equals fraction numerator left parenthesis S R right parenthesis left parenthesis x right parenthesis over denominator 2 end fraction space space space space space space space space space space space equals space fraction numerator left parenthesis P Q right parenthesis left parenthesis x right parenthesis over denominator 2 end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 2 right parenthesis

∵ SR = PQ (opposite sides of parallelogram are equal)
Therefore, either the farmer should sow wheat in ΔAPQ and pulses in the other two triangles APS and AQR or pulses in ΔAPQ and wheat in the other two triangles APS and AQR.

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8. The diagonals of a parallelogram ABCD intersect at a point O. Through O, a line is drawn to intersect AD at P and BC at Q. Show that PQ divides the parallelogram into two parts of equal area.

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9. Prove that of all parallelograms of which the sides are given, the parallelogram which is a rectangle, has the greatest area.

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10. In the figure, diagonals AC and BD of a trapezium ABCD with AB || CD intersect each other at O. Show that ar(Δ AOD) = ar(Δ BOC).

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