State whether the given statement is true or false:

In the given figure, D and E are two points on side BC of $Δ A B C$ such that $B D = D E = E C$ , then the area of triangle ABD is equal to that of ADE and AEC.

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True

False

answer is A.

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Detailed Solution

Given a triangle

Δ A B C

in which D and E are points on the side BC such that

B D = D E = E C

We have to check whether the area of triangle ABD is equal to that of ADE and AEC or not.
It is given that

B D = D E = E C

.
This means the length of the base of triangles ABD, ADE and AEC is of the same length and they share a common vertex at A. … (1)
In the figure, the height or the altitude is AL. This is the height for all three triangles, ABD, ADE and AEC.
Area of a triangle =

\frac{1}{2} \times Base \times Height

So, area of

ΔABD = \frac{1}{2} \times BD \times AL

.
Area of

ΔADE = \frac{1}{2} \times DE \times AL

Area of

ΔAEC = \frac{1}{2} \times CE \times AL

Since the bases are already equal and height is common, the area of triangles ABD, ADE and AEC are equal.
So, we can conclude that the area of triangle ABD is equal to that of ADE and AEC.
Therefore, the given statement is true.
Hence, option 1 is the correct choice.

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