ABCD is a parallelogram. M is the mid-point of AB and P is a point on diagonal BD such that BP = 1 4 BD . MP produced meets BC at N . Prove that N is a mid-point of BC .

Let us draw the diagonal BD as shown below. The diagonal AC and BD cuts at point X . We know that the diagonals of a parallelogram bisect each other. Therefore, DX = BX and AX = CX Given, BP = 1 4 BD BP = 1 4 × 2 BX BP = 1 2 BX Therefore, P is the mid point of BX . For △ BAX , MP ∥ AX or MN ∥ AC ANd in △ BCX , P is the mid point of BX and PN ∥ CX Therefore, N is the mid-point of BC .

ABCD is a parallelogram. M is the mid-point of AB and P is a point on diagonal BD such that BP = 1 4 BD . MP produced meets BC at N . Prove that N is a mid-point of BC .

Let us draw the diagonal BD as shown below. The diagonal AC and BD cuts at point X . We know that the diagonals of a parallelogram bisect each other. Therefore, DX = BX and AX = CX Given, BP = 1 4 BD BP = 1 4 × 2 BX BP = 1 2 BX Therefore, P is the mid point of BX . For △ BAX , MP ∥ AX or MN ∥ AC ANd in △ BCX , P is the mid point of BX and PN ∥ CX Therefore, N is the mid-point of BC .

E M B I B E

ICSE MATHEMATICS CLASS 9>Chapter 10 - Mid-Point Theorem>EXERCISE 10>Q 12

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$ABCD$ is a parallelogram. $M$ is the mid-point of $AB$ and $P$ is a point on diagonal $BD$ such that $BP = \frac{1}{4} BD$ . $MP$ produced meets $BC$ at $N$ . Prove that $N$ is a mid-point of $BC$ .

Important Questions on Mid-Point Theorem

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ICSE MATHEMATICS CLASS 9>Chapter 10 - Mid-Point Theorem>EXERCISE 10>Q 12

$ABCD$ is a parallelogram. $M$ is the mid-point of $AB$ and $P$ is a point on diagonal $BD$ such that $BP = \frac{1}{4} BD$ . $MP$ produced meets $BC$ at $N$ . $MN = \frac{1}{2} AC$

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ICSE MATHEMATICS CLASS 9>Chapter 10 - Mid-Point Theorem>EXERCISE 10>Q 13

In $△ ABC, AM$ is a median and $N$ is the mid-points of $AM$ . $BN$ produced meets $AC$ at $P$ . Prove that $AP = \frac{1}{3} AC$ .

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ICSE MATHEMATICS CLASS 9>Chapter 10 - Mid-Point Theorem>EXERCISE 10>Q 14

$ABCD$ is a rectangle. $P, Q, R$ and $S$ are mid-points of sides of the rectangle as shown in the given figure. Prove that $PQRS$ is a rhombus.

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ICSE MATHEMATICS CLASS 9>Chapter 10 - Mid-Point Theorem>EXERCISE 10>Q 15

A B C D

is a parallelogram,

E

and

F

are the midpoints of

A B

and

C D

respectively.

G H

is any line intersecting

A D, E F

and

B C

at

G, P

and

H

respectively. Prove that

G P = P H

.

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ICSE MATHEMATICS CLASS 9>Chapter 10 - Mid-Point Theorem>EXERCISE 10>Q 16

$ABC$ is a right-angled triangle with hypotenuse $AC = 13 cm$ and side $AB = 5 cm$ . Perpendiculars are drawn from mid-point $M$ of $AC$ to $AB$ and $B C$ . What is the perimeter of the resulting quadrilateral?

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ICSE MATHEMATICS CLASS 9>Chapter 10 - Mid-Point Theorem>EXERCISE 10>Q 17

In the quadrilateral $ABCD, AD ∥ BC . P$ and $Q$ are mid-points of $AB$ and $AC$ . Prove that $R$ is the mid-point of $DC$ .

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9th ICSE

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ICSE MATHEMATICS CLASS 9>Chapter 10 - Mid-Point Theorem>EXERCISE 10>Q 17

In the quadrilateral $ABCD, AD ∥ BC . P$ and $Q$ are mid-points of $AB$ and $AC$ . Prove that $AD + BC = 2 PR$ .

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ICSE MATHEMATICS CLASS 9>Chapter 10 - Mid-Point Theorem>EXERCISE 10>Q 18

In parallelogram $PQRS, M$ is the mid-point of $PQ$ , $PT$ drawn parallel to $MR$ meets $SR$ at $N$ and $QR$ produced at $T$ . Prove that $PS = \frac{1}{2} QT$ .