E and F are points on diagonal AC of a parallelogram ABCD such that AE = CF . Show that BFDE is a parallelogram.

Given a parallelogram ABCD and E and F are points on diagonal AC of parallelogram ABCD such that AE = CF ABCD is a parallelogram. Since, diagonals of parallelogram bisect each other. ∴ OD = OB ...(i) OA = OC ...(ii) AE = CF ...(iii) (given) Subtracting (iii) from (ii), we get OA - AE = OC - CF ⇒ OE = OF ...(iv) ∴ BFDE is a parallelogram. [ ∵ OD = OB and OE = OF ] Hence, proved.

E M B I B E

NCERT Exemplar Mathematics - Class 9>Chapter 8 - Quadrilaterals>Exercise>Q 4

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9th CBSE

IMPORTANT

Earn 100

$ABCD$ is a rhombus in which altitude from $D$ to side $AB$ bisects $AB$ . Find the angles of the rhombus.

Important Questions on Quadrilaterals

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IMPORTANT

NCERT Exemplar Mathematics - Class 9>Chapter 8 - Quadrilaterals>Exercise>Q 5

E

and

F

are points on diagonal

\Rightarrow ∠ ABC = 180 ° - 60 ° = 120 °

of a parallelogram

ABCD

such that

AE = CF

. Show that

BFDE

is a parallelogram.

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9th CBSE

IMPORTANT

NCERT Exemplar Mathematics - Class 9>Chapter 8 - Quadrilaterals>Exercise>Q 6

E

is the midpoint of the side

AD

of the trapezium

ABCD

with

AB ∥ DC

.

A

line through

E

drawn parallel to

AB

intersect

BC

at

F

. Show that

F

is the midpoint of

BC

.

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IMPORTANT

NCERT Exemplar Mathematics - Class 9>Chapter 8 - Quadrilaterals>Exercise>Q 7

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 $BC = \frac{1}{2} QR$ .

Question Image

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IMPORTANT

NCERT Exemplar Mathematics - Class 9>Chapter 8 - Quadrilaterals>Exercise>Q 8

D, E

and

F

are the midpoints of the sides

BC, CA

and

AB

, respectively of an equilateral triangle

ABC

. Show that

△ DEF

is also an equilateral triangle.

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IMPORTANT

NCERT Exemplar Mathematics - Class 9>Chapter 8 - Quadrilaterals>Exercise>Q 9

Points

P

and

Q

have been taken on opposite sides

AB

and

CD

, respectively of a parallelogram

ABCD

such that

AP = CQ

. Show that

AC

and

PQ

bisect each other.

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9th CBSE

IMPORTANT

NCERT Exemplar Mathematics - Class 9>Chapter 8 - Quadrilaterals>Exercise>Q 10

In the given figure,

P

is the midpoint of side

BC

of a parallelogram

ABCD

such that

∠ BAP = ∠ DAP

. Prove that

AD = 2 CD

.

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IMPORTANT

NCERT Exemplar Mathematics - Class 9>Chapter 8 - Quadrilaterals>Exercise>Q 1

A square is inscribed in an isosceles right triangle so that the square and the triangle have one angle common. Show that the vertex of the square opposite the vertex of the common angle bisects the hypotenuse.

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IMPORTANT

NCERT Exemplar Mathematics - Class 9>Chapter 8 - Quadrilaterals>Exercise>Q 3