Question

in triangle ABC, if DE║BC and DE:BC = 5:8 , Then find the ratios of areas trapezium BCED and triangle ABC

Answer 1

Given:

In triangle ABC,

DE║BC

and

DE:BC = 5:8

To find:

The ratios of areas trapezium BCED and triangle ABC

Solution:

In ΔADE and ΔABC, we have

∠DAE = ∠BAC ...... [common angles]

∠ADE = ∠ABC ...... [∵ DE // BC, corresponding angles]

∴ ΔADE ~ ΔABC ..... [By AA Similarity]

We know that → the ratio of the areas of the two similar triangles is equal to the ratio of the squares of their corresponding sides.

So, based on this, we can say that,

$\frac{Area(\triangle ADE)}{Area (\triangle ABC)} = \frac{(DE)^2}{(BC)^2}$

substituting DE:BC = 5:8

$\implies \frac{Area(\triangle ADE)}{Area (\triangle ABC)} = \frac{(5)^2}{(8)^2}$

$\implies \bold{\frac{Area(\triangle ADE)}{Area (\triangle ABC)} = \frac{25}{64}}$ ...... (i)

Now,

$\frac{Area (Trapezium \:BCED)}{Area (\triangle ABC)}$

= $\frac{Area (\triangle ABC) \:-\:Area (\triangle ADE) }{Area (\triangle ABC)}$

substituting the values from (i)

= $\frac{64 \:-\:25 }{64}$

= $\bold{\frac{39 }{64}}$

Thus, the ratios of areas of trapezium BCED and triangle ABC is → $\underline{\bold {\frac{39}{64} }}$.

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