In the given figure, A B C D is a parallelogram in which P is the mid-point of D C and Q is a point on A C , such that C Q = 1 4 A C . Also, P Q when produced meets B C at R . Prove that R is the mid-point of B C .

Figure is given as follows. A B C D is a parallelogram, where P is the midpoint of D C and Q is a point on A C such that C Q = 1 4 A C . P Q produced meets B C at R . We need to prove that R is a midpoint of B C . Let us join B D to meet A C at O . It is given that A B C D is a parallelogram. Therefore, O C = 1 2 A C (Because diagonals of a parallelogram bisect each other) Also, C Q = 1 4 A C Therefore, C Q = 1 2 O C In △ D C O , P and Q are the midpoints of C D and O C respectively. Therefore, we get P Q ∥ D O (By midpoint theorem) Also, in △ C O B , Q is the midpoint of O C and Q R ∥ O B Therefore, R is a midpoint of B C . (By converse of midpoint theorem) Hence, proved.

E M B I B E

All in One Mathematics>Chapter 8 - Quadrilaterals>Exam Practice>Q 40

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IMPORTANT

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The diagonals of a quadrilateral $A B C D$ are perpendicular, show that quadrilateral formed by joining the mid-points of its sides, is rectangle.

Important Questions on Quadrilaterals

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IMPORTANT

All in One Mathematics>Chapter 8 - Quadrilaterals>Exam Practice>Q 41

A diagonal of a parallelogram bisects one of its angles. Show that it is a rhombus.

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9th CBSE

IMPORTANT

All in One Mathematics>Chapter 8 - Quadrilaterals>Exam Practice>Q 42

Prove that the diagonal of a parallelogram divides it into two congruent triangles.

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All in One Mathematics>Chapter 8 - Quadrilaterals>Exam Practice>Q 43

Prove that the line joining mid-points of the diagonals of a trapezium is parallel to the parallel sides of the trapezium.

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All in One Mathematics>Chapter 8 - Quadrilaterals>Exam Practice>Q 44

In the given figure, $B E ⊥ A C$ . $A D$ is any lire from $A$ to $B C$ intersecting $B E$ at $H . P, Q$ and $R$ are mid-points of $A H, A B$ and $B C$ respectively, then prove that $∠ P Q R = 90 °$ .

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All in One Mathematics>Chapter 8 - Quadrilaterals>Exam Practice>Q 45

The angle between two altitudes of a parallelogram through the vertex of an obtuse angle of the parallelogram is $60 °$ . Find the angles of the parallelogram.

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All in One Mathematics>Chapter 8 - Quadrilaterals>Exam Practice>Q 46

P

and

Q

are points on opposite sides

A D

and

B C

of a parallelogram

A B C D

such that

P Q

passes through the point of intersection

O

of its diagonals

AC

and

BD

. Show that

PQ

is bisected at

O

.

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IMPORTANT

All in One Mathematics>Chapter 8 - Quadrilaterals>Exam Practice>Q 47

Through $A, B$ and $C,$ lines $RQ, PR$ and $QP$ have been drawn, respectively parallel to sides $BC, CA$ and $AB$ of a $△ ABC$ as shown in the given figure. Show that $BC = \frac{1}{2} QR$ .

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9th CBSE

IMPORTANT

All in One Mathematics>Chapter 8 - Quadrilaterals>Exam Practice>Q 48

In the given figure, $A B C D$ is a parallelogram in which $P$ is the mid-point of $D C$ and $Q$ is a point on $A C$ , such that $C Q = \frac{1}{4} A C$ . Also, $P Q$ when produced meets $B C$ at $R$ . Prove that $R$ is the mid-point of $B C$ .