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Constructions

Question
CBSEENMA9002891
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In the given figure, ΔABC is right angled at B, and BD is its median. E is the mid-point of BD. If AB = 6 cm, AC = 10 cm, calculate area of ΔBEC.

Solution

In right triangle ABC,
AB2 + BC2 = AC2
| By Pythagoras theorem ⇒    62 + BC2 = 102
⇒    BC = 8 cm
therefore space space ar left parenthesis increment ABC right parenthesis equals 1 half cross times space Base space cross times space Corresponding space altitude space space space space space space space space space space space space space space space space space space equals space 1 half cross times BC cross times AB space space space space space space space space space space space space space space space space space space space equals space 1 half cross times 8 cross times 6 space space space space space space space space space space space space space space space space space space space equals space 24 space cm squared 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
∵ BD is a median of ΔABC
therefore space space space ar space left parenthesis increment ABD right parenthesis space equals space ar left parenthesis increment BCD right parenthesis equals 1 half ar left parenthesis increment ABC right parenthesis space space space space space space space space space space space space space space equals space 1 half cross times 24 space space space space 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 space equals space 12 space cm squared space

∵ A median of a triangle divides it into two triangles of equal areas ⇒ ar(ΔBCD) = 12 cm2    ...(2)
∵ E is the mid-point of BD ∴ CE is a median of ΔBCD
therefore space ar left parenthesis increment BEC right parenthesis space equals space ar space left parenthesis increment CED right parenthesis equals 1 half ar space left parenthesis increment BCD right parenthesis
∵ A median of a triangle divides it into two triangles of equal areas
space space space space space space space space space space space space space equals space 1 half cross times 12 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 vertical line space From space left parenthesis 2 right parenthesis space space space space space space space space space space space space space space equals space 6 space space cm squared space space rightwards double arrow space space space space space ar left parenthesis increment BEC right parenthesis equals 6 space cm squared

Some More Questions From Constructions Chapter

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

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.

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