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Standard IX

Mathematics

ASA Criteria for Congruency

ABCD is a par...

Question

ABCD is a parallelogram. Two lines l and m are parallel to AD. Line I meets AB and CD at P and S respectively. Line m meets AB and CD at Q and R respectively. X is any point on CD between R and S. If ar( $Δ$ DPX) $+$ ar( $Δ$ CQX) $=$ k ar(ABCD), find k.

\frac{2}{3}

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\frac{3}{2}

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\frac{1}{2}

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\frac{1}{3}

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Solution

The correct option is D $\frac{1}{2}$
In parallelogram $A B C D$ , Opposite sides are congruent
$A D = B C$
and $∵ A D | | P S | | Q R | | B C$
$∴ P S = A D = Q R = B C$
$a r (△ D P X) + a r (△ C Q X) = \frac{1}{2} P S \times D X + \frac{1}{2} Q R \times C X = \frac{1}{2} \times A D (D X + X C) = \frac{1}{2} A D \times C D = \frac{1}{2} a r (A B C D)$
$∴ k = \frac{1}{2}$

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Similar questions

Q. A point P is taken on side CD of a parallelogram ABCD and CD is produced to Q making DQ = CP. The line through Q parallel to AD meets BP produced at S and AD is produced to meet BS at R. Prove that ARSQ is a parallelogram.

Q. ABCD is a parallelogram . AB is divided at P and CD at Q so that AP : PB = 3 : 2 and CQ : QD = 4 : 1 . If PQ meets AC at R then prove that AR =

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Q. Let

- - \to A B = 3^i -^j +^k

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are

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and

- 9^j + 2^k

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and a point

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such that

- - \to P Q

is perpendicular to the vectors

- - \to A B

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$- - \to A B = 3^i -^j +^k a n d - - \to C D = - 3^i + 2^j + 4^k$ are two vectors.The position vectors of the points A and C are $6^i + 7^j + 4^k a n d - 9^i + 2^j$ ,respectively. Find the position of vector of a point P on the line AB and a point Q on the line CD such that $- - \to P Q$ is perpendicular to $- - \to A B a n d - - \to C D$ both.

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ABCD is a parallelogram. Two lines l and m are parallel to AD. Line I meets AB and CD at P and S respectively. Line m meets AB and CD at Q and R respectively. X is any point on CD between R and S. If ar(ΔDPX)+ar(ΔCQX)=k ar(ABCD), find k.

The correct option is D 12In parallelogram ABCD, Opposite sides are congruentAD=BCand ∵AD||PS||QR||BC∴PS=AD=QR=BCar(△DPX)+ar(△CQX)=12PS×DX+12QR×CX=12×AD(DX+XC)=12AD×CD=12ar(ABCD)∴k=12

ABCD is a parallelogram. Two lines l and m are parallel to AD. Line I meets AB and CD at P and S respectively. Line m meets AB and CD at Q and R respectively. X is any point on CD between R and S. If ar( $Δ$ DPX) $+$ ar( $Δ$ CQX) $=$ k ar(ABCD), find k.