ABCD is parallelogram- P is a point on AD such that AP -13AD and Q is a point on BC such that CQ -13 BC- Prove that AQCP is a parallelogram-

ABCD is a parallelogram. P is a point on AD AP=13AD #160; #160; #160; #160; #160; #160; #160;[ Given ] #160; #16 ...

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Byju's Answer

Standard X

Mathematics

Basic Proportionality Theorem

ABCD is paral...

Question

ABCD is parallelogram. P is a point on AD such that AP $= \frac{1}{3}$ AD and Q is a point on BC such that CQ $= \frac{1}{3}$ BC. Prove that AQCP is a parallelogram.

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Solution

$A B C D$ is a parallelogram. $P$ is a point on $A D$
$A P = \frac{1}{3} A D$ [ Given ] ----- ( 1 )

$C Q = \frac{1}{3} B C$ [ Given ] --- ( 2 )

$\Rightarrow$ $A D = B C$ and $A D ∥ B C$

$\Rightarrow$ $\frac{1}{3} A D = \frac{1}{2} B C$ and $A D ∥ B C$

$\Rightarrow$ $A P = C Q$ and $A P = C Q$ [ From ( 1 ) and ( 2 ) ]

Thus, $A Q C P$ is a quadrilateral such that one pair of opposite side

$A P$ and $C Q$ are parallel and equal.

$∴$ $A Q C P$ is a parallelogram.

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