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Select the correct option from the following:

ABCD is a parallelogram. If P and Q are points on AD and BC respectively such that $A P = 13 A D$ and $C Q = 13 B C$ . Then ACPQ is a

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Parallelogram

Kite

Trapezoid

Square

answer is A.

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Detailed Solution

We are given that,

A P = 13 A D

and

C Q = 13 B C

.
Consider the figure given below.
Question Image

Since opposite sides and angles of a parallelogram are equal, we have,

∠ B = ∠ D, A D = B C,

AB = CD.
Also, AD || BC and AB || DC.
AD = BC

⇒ 13 A D = 13 B C ⇒ A P = C Q

... (i)
Consider

Δ D P C

and

Δ B Q A

.
AB = DC

∠ B = ∠ D

DP = BQ (

A P = 13 A D

and

C Q = 13 B C

)
Hence, by SAS rule for congruency,

Δ D P C ≅ Δ B Q A

.
Therefore, by CPCT
PC = QA ... (ii)
From equations (i) and (ii),
AP = CQ,
PC = QA.
Since in quadrilateral AQCP, two opposite sides are equal, it is a parallelogram. Therefore, option 1 is the correct answer.

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