Theorem 1: Parallelograms

Q.

PQ and RS are two equal and parallel line segments.
Any point M not lying on PQ or RS is joined to Q and S and lines through P parallel to QM and through R parallel to SM meet at N . Prove that line segments MN and PQ are equal and parallel to each other.

Q. X and Y are points on the side LN of the triangle LMN such that LX = XY = YN. Through X, a line is drawn parallel to LM to meet MN at Z (See figure). Then:

1. $Area\left(\Delta LZY\right)=Area\left(MZYX\right)$
2. $Area\left(\Delta NMX\right)=Area\left(MZYX\right)$
3. $Noneofthese$
4. $Area\left(\Delta LZN\right)=Area\left(MZYX\right)$

Q.

In ABC , Given that DE//BC , D is the midpoint of AB and E is a midpoint of AC. The ratio AE : EC is ____.

1. 1 : 1

2. 1 : 3

3. 1 : 2

4. 2 : 1

Q. ABCD is a parallelogram, E and F are the midpoints of AB and CD respectively then the ratio of areas of AEFD and EBCF is ______ .

1. 1:3
2. 1:1
3. 1:2
4. 2:1

Q.

Parallelograms an equal bases and between the same parallels are equal in area

1. False

2. True

Q.

If parallelogram ABCD and rectangle ABEM are of equal area, then:

1. Perimeter of AMEB = Perimeter of ADCB

2. Perimeter of AMEB Perimeter of ADCB

3. Perimeter of AMEB Perimeter of ADCB

4. Perimeter of AMEB = Perimeter of ADCB

Q. Question 9 (ii)
In parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP=BQ (see the given figure) Show that:
AP = CQ

Q. In figure, if points P, Q, R, S are on the sides of parallelogram such that AP = BQ = CR = DS then prove that PQRS is a parallelogram

Q.

$D$ and $E$ are points on sides $AB$ and $AC$ respectively of $∆ABC$ such that $ar\left(DBC\right)=ar\left(EBC\right)$ . Prove that $DE\parallel BC$

Q. Find the perimeter of the trapezium WXYZ if WX = 7m, YZ = 13m and ZW = 8m.

1. 30m
2. 39m
3. 32m
4. 38m

Q.

Two photos are similar. The ratio of the corresponding side lengths is $3:4$. What is the ratio of the areas?

Q.

The playing surfaces of two foosball tables are similar. The ratio of the corresponding side lengths is $10:7$. What is the ratio of the areas?

Q.

In the given figure, PQRS is a rectangle of area $35c{m}^2$. Find the area of $\Delta EFT.$

1. $35c{m}^2$
2. $70c{m}^2$
3. $17.5c{m}^2$
4. $60c{m}^2$

Q. If two parallelograms have equal bases and are between the same parallels, then the ratio of their areas is .

1. 1 : 2
2. 1 : 1
3. 2 : 1
4. 3 : 1

Q. Two triangles have the same base and equal areas have their vertex lying on the same side of their base. The line joining the two apexes will be _______ to their base.

1. double
2. intersecting
3. parallel
4. equal

Q.

In the given figure, ABQP and ABCD are two parallelograms on the same base AB. $\frac{\text{Area of parallelogram ABCD}}{\text{Area of triangle PAB}}$ is equal to:

__

Q.

In the given figure area of $\Delta DAC$ is $60c{m}^2$. Find the area of parallelogram ABCD.

[2 Marks]

Q. P, Q, R, S are respectively the midpoints of the sides AB, BC, CD and DA of ||gm ABCD. Show that PQRS is a parallelogram and also show that $ar\left(||gmPQRS\right)=\frac{1}{2}ar\left(||gmABCD\right)$.

Q.

Find the value of the ratio (red to blue) of the areas of the similar figures.

Q. Question 2
The diagonals of a parallelogram ABCD intersect at a point O. Through O, a line is drawn to intersect AD at P and BC at Q. Show that PQ divides the parallelogram into two parts of equal area.

Q. ABCD is a parallelogram and AP and CQ are perpendiculars from vertices A and C on diagonal BD( See the given figure). Show that
AP = CQ

Q. The side AB of a parallelogram ABCD is produced to any point P. A line through A and parallel to CP meets CB produced at Q and then parallelogram PBQR is completed. Show that
ar(ABCD) = ar(PBQR).
[Hint : Join AC and PQ. Now compare ar(ACQ) and ar(APQ).]

Q. ABCD is a parallelogram and AP and CQ are perpendiculars from vertices A and C on diagonal BD( See the given figure). Show that

$\Delta APB\cong \Delta CQD$

Q. ABCD is a parallelogram and AP and CQ are perpendiculars from vertices A and C on diagonal BD( See the given figure). Show that

$\Delta APB\cong \Delta CQD$

Q. If a triangle and parallelogram are on the same base and between the same parallels, then the ratio of area of the triangle to that of area of the parallelogram is ___.

1. 1:2
2. 3:1
3. 1:4
4. 1:3

Q. In parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ (see the given figure).


Identify the incorrect statement from the following.

1. 
   $\Delta APD\cong \Delta CQB$
2. 
   AP = CQ
3. 
   $\Delta AQB\cong \Delta CPD$
4. 
   AD = CQ

Q.

The area of parallelogram ABCD is:

1. AB×BM
2. BC×BN
3. DC×DL
4. AD×BM

Q.

Which among the following is correct?
Parallelograms on the same base and between the same set of parallel lines:

1. have unequal areas

2. have the same perimeter

3. may have equal or unequal areas

4. have equal areas

Q. Two triangles have the same base and equal areas have their vertex lying on the same side of their base. The line joining the two apexes will be _______ to their base.

1. equal
2. double
3. parallel
4. intersecting

Q. The area of a parallelogram whose diagonal is 6.8 cm and the perpendicular distance of this diagonal from an opposite vertex is 7.5 cm is

1. 25.5 $c{m}^2$
2. 11.9 $c{m}^2$
3. 12.5 $c{m}^2$
4. 51 $c{m}^2$