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

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71 : 21

12 : 11

31 : 35

4 : 7

answer is C.

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Detailed Solution

Given that in 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 + 5 = y + 1 . . . . . . . . . . (1)

\Rightarrow 3 x - 4 = y + 5 . . . . . . . . . (2)

Subtracting Equation 1 from equation 2,

(3 x -) 4 - (2 x + 5) = y + 5 - (y + 1)

\Rightarrow 3 x - 4 - 2 x - 5 = y + 5 - y - 1

\Rightarrow x - 4 - 5 = 4

\Rightarrow x - 9 = 4

\Rightarrow x = 4

+ 9

\Rightarrow x = 13

Put value of x in equation 1,

2 (13) + 5 - 1 = y

\Rightarrow 26 + 5 - 1 = y

\Rightarrow y = 30

Hence, x =13 and y =30.

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

\Rightarrow \frac{AB}{BC} = \frac{2 (13) + 5}{3 (13) - 4}

\Rightarrow \frac{AB}{BC} = \frac{26 + 5}{39 - 4}

\Rightarrow \frac{AB}{BC} = \frac{31}{35}

Therefore, the ratio,

AB : BC = 31 : 35

.
Hence, option 3 is correct.

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