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  • In Fig. , if DE || BC, find the ratio of ar (ADE) and ar (DECB).

In Fig., if DE || BC, find the ratio of ar (ADE) and ar (DECB).

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Answer : 1: 3

Given:- DE || BC and DE = 6 cm, BC = 12 cm

In \Delta ABC \; and\; \Delta ADE\angle ABC = \angle ADE  \text{ (corresponding angle) }
\angle A = \angle A  \text{ (common angle) }

As we know if two angles of one triangle are equal to the two angles of another triangle, then the two triangles are similar by AA similarity criterion.

\therefore \; \; \; \; \; \Delta ABC \sim \Delta ADE

Then

\frac{ar\left ( \Delta ADE \right )}{ar\left ( \Delta ABC \right )}=\left ( \frac{DE}{BC} \right )^{2}=\left ( \frac{6}{12} \right )^{2}=\left ( \frac{1}{2} \right )^{2}=\frac{1}{4}

Let \; ar\left ( \Delta ADE \right )=\text {k then ar}\left ( \Delta ABC \right )=4k

Now\; ar(\Delta ECB) = ar(\Delta ABC) = ar(\Delta ADE) = 4k - k = 3k

\therefore Required \; ratio = ar(ADE) : ar (DECB)

\\k:3k\\1:3

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