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Q.

In the figure, ABCDE is a pentagon with BE||CD and BC||DE. BC is perpendicular to CD. AB = 5cm, AE = 5cm, BE = 7cm, BC = x-y and CD = x + y. If the perimeter of ABCDE is 27cm. find the value of x and y, given x, y ≠ 0.

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a

x = -6, y = 1

b

x = 6, y = 1

c

x = 6, y = -1

d

x = 5, y = -1

answer is B.

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Detailed Solution

Given that, in pentagon ABCDE, BE || CD and BC || DE & BC ⊥ CD AB = 5cm, AE = 5cm, BE = 7cm, BC = x - y, CD = x + y.
Also given, perimeter of ABCDE = 27 cm.

Question Image

As, BE || CD and BC || DE, the corresponding angles are equal.
Since, BC ⊥ CD,
∴ ∠BCD = ∠CDE = ∠DEB = ∠EBC = 90°
Hence, BCDE is a rectangle.
In rectangle BCDE, BE = 7cm, BC = x - y, CD = x + y
As we know that, opposite sides of a rectangle are parallel and equal.
∴ BE = CD = 7cm
∴ x + y = 7 ………(1)
And BC = DE
∴ BC = DE = x – y ………(2)
We have, perimeter of ABCDE = 27 cm,
AB = 5cm, AE = 5cm, BC = DE = x - y, CD = 7cm and CD = x + y.
We know that, Perimeter of pentagon is the sum of all of its sides.
Perimeter of pentagon ABCDE = AB + BC + CD + DE + EA

\Rightarrow

27 = 5 + (x – y) + 7 + (x – y) + 5

\Rightarrow

27 = 17 + 2x – 2y

\Rightarrow

27 - 17 = 2x – 2y

\Rightarrow

10 = 2(x – y)

\Rightarrow

(x – y) = 10/2

\Rightarrow

x – y = 5 ………(3)
Add eq (1) and (3).
x + y = 7
x – y = 5
----------------
2x = 12
----------------

\Rightarrow

x = 12/2

\Rightarrow

x = 6
Substitute the value of x = 6 in eq (1) and find y.
x + y = 7

\Rightarrow

6 + y = 7

\Rightarrow

y = 7 - 6

\Rightarrow

y = 1
Hence, x = 6 and y = 1.
The correct option is (2).

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