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Constructions

Question
CBSEENMA9002869
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In a triangle ABC, E is the midpoint of median AD. Show that a r left parenthesis increment B E D right parenthesis space equals 1 fourth a r left parenthesis increment A B C right parenthesis

Solution
Given: In a triangle ABC, E is the midpoint of median AD.
 To space Prove space colon space ar left parenthesis increment BED right parenthesis equals 1 fourth ar left parenthesis increment ABC right parenthesis.
Proof: In ΔABC,

∵ AD is a median.
therefore space space ar left parenthesis increment ABD right parenthesis equals ar left parenthesis increment ACD right parenthesis equals 1 half ar left parenthesis increment ABC right parenthesis

(1)

∵ A median of a triangle divides it into two triangles of equal areas.
In ΔABD,
∵ BE is a median.

therefore space space ar left parenthesis increment BED right parenthesis equals ar left parenthesis increment BEA right parenthesis equals 1 half ar left parenthesis increment ABD right parenthesis
∵ A median of a triangle divides it into two triangles of equal areas
rightwards double arrow space space ar left parenthesis increment BED right parenthesis equals 1 half ar left parenthesis increment ABD right parenthesis space space space space space space space space space space space space space space equals 1 half.1 half space ar left parenthesis increment ABC right parenthesis space space space space space space space space space space space space space vertical line space From space left parenthesis 1 right parenthesis space space space space space space space space space space space space space space equals space 1 fourth ar space left parenthesis increment ABC right parenthesis.

Some More Questions From Constructions Chapter

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).

Parallelograms on the same base and between same parallels are equal in area. Prove this.

ABCD is a quadrilateral and BD is one of its diagonals as shown in figure. Show that ABCD is a parallelogram and find its area.

In the given figure, AB | | DC. Show that ar(BDE) = ar(ACED).

 In the given figure, ABED is a parallelogram in which DE = EC. Show that area (ABF) = area (BEC)

In the following figure, ABCD is a parallelogram and EFCD is a rectangle. Also, AL ⊥ DC. Prove that
(i) ar(ABCD) = ar(EFCD)
(ii) ar(ABCD) = DC x AL.

If a triangle and a parallelogram are on the same base and between the same parallels, then prove that the area of the triangle is equal to half the area of the parallelogram.

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