Question

In the given figure, ΔABC is right angled at A and AD is its median. BA is produced to E such that BA = AE. ED is joined. If AB = 6 cm, BC = 10 cm, find ar(ΔBED).

Solution

In right triangle BAC,
AB²+ AC² = BC²| By Pythagoras Theorem
⇒ 6² + AC² = 10² ⇒ AC = 8 cm
$therefore space space space ar left parenthesis increment ABC right parenthesis equals 1 half cross times space Base space cross times thin space Corresponding space altitude space space space space space space space space space space space equals space 1 half cross times AB cross times AC space space space space space space space space space space space equals space 1 half cross times 6 cross times 8 space space space space space space space space space space space space equals space 24 space cm squared space space space space space space space space space space space space space space space space space space space space... left parenthesis 1 right parenthesis$

∵ AD is a median of ΔABC ∴ ar(ΔABD) = ar(ΔACD)
∵ A median of a triangle divides it into two triangles of equal areas
$equals space 1 half space ar left parenthesis ABC right parenthesis equals space 1 half straight x space 24 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space vertical line space From space left parenthesis 1 right parenthesis equals space space 12 space cm squared space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 2 right parenthesis space space space$
∵ BA = AE ∴ A is the mid-point of BE ∴ DA is a median of ΔBDE
$therefore space space ar space left parenthesis increment ABD right parenthesis equals 1 half ar left parenthesis increment BED right parenthesis$
∵ A median of a triangle divides it into two triangles of equal areas
$rightwards double arrow space space space space space ar left parenthesis increment ABD right parenthesis equals 1 half ar left parenthesis increment BED right parenthesis rightwards double arrow space space space space space 12 equals 1 half ar left parenthesis increment BED right parenthesis space space space space space space space vertical line space From space left parenthesis 3 right parenthesis rightwards double arrow space space space space space ar left parenthesis increment BED right parenthesis equals 24 space cm squared$

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Constructions

In the given figure, ΔABC is right angled at A and AD is its median. BA is produced to E such that BA = AE. ED is joined. If AB = 6 cm, BC = 10 cm, find ar(ΔBED).

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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