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In a parallelogram $ABCD$ , if $AB = 2 x + 6, CD = y + 1, AD = y + 5$ and $BC = 3 x - 4$ , what is the ratio of $AB$ and $BC$ ?

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71 : 21

12 : 11

33 : 38

4 : 7

answer is C.

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Detailed Solution

Given that a parallelogram

ABCD

AB = 2 x + 5, CD = y + 1, AD = y + 5

and

BC = 3 x - 4

We have to find the ratio of

AB

and

BC

.
In parallelogram, opposite sides are parallel and equal.

∴ AB = CD

and

AD = BC

\Rightarrow 2 x + 6 = y + 1 - - - - 1

\Rightarrow 3 x - 4 = y + 5 - - - - 2

Subtracting Equation 1 from equation 2,

(3 x - 4) - (2 x + 6) = y + 5 - (

y + 1

)

\Rightarrow 3 x - 4 - 2 x - 6 = y + 5 -

y - 1

\Rightarrow x - 10 = 4

\Rightarrow x = 4 + 10

\Rightarrow x = 14

Put the value of x in 1.

2 (14) + 5 - 1 = y

\Rightarrow 28 + 5 - 1 = y

\Rightarrow y = 32

Thus,

\frac{AB}{BC} = \frac{2 x + 5}{3 x - 4}

\Rightarrow \frac{AB}{BC} = \frac{2 (14) + 5}{3 (14) - 4}

\Rightarrow \frac{AB}{BC} = \frac{28 + 5}{42 - 4}

\Rightarrow \frac{AB}{BC} = \frac{33}{38}

Therefore ratio of

AB : BC = 33 : 38

.
Hence option 3 is correct.

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