Properties of Diagonal of a Parallelogram

Q.

Prove that the diagonals of a parallelogram bisect each other.

Q. Show that the diagonals of a parallelogram divide it into four triangles of equal area. [3 MARKS]

Q.

The diagonals of a parallelogram ABCD intersect at O. If $BOC={90}^{\circ }and\angle BDC={50}^{\circ }$ then $\angle OAB=$

1. ${40}^{\circ }$

2. ${50}^{\circ }$

3. ${10}^{\circ }$

4. ${90}^{\circ }$

Q.

The diagonals AC and BD of a parallelogram ABCD intersect each other at the point O such that $\angle DAC={30}^{\circ }and\angle AOB={70}^{\circ }$. Then, $\angle DBC=$ ?

(a) ${40}^{\circ }$

(b) ${35}^{\circ }$

(c) ${45}^{\circ }$

(d) ${50}^{\circ }$

Q.

In the given figure, ABCD is a parallelogram, M is the midpoint of BD and BD bisects $\angle B$ as well as $\angle D$. Then, $\angle AMB=$ ?

(a) ${45}^{\circ }$

(b) ${60}^{\circ }$

(c) ${90}^{\circ }$

(d) ${30}^{\circ }$

Q.

$P$ and $Q$ are the points of trisection of the diagonal $BD$ of the parallelogram $ABCD$, Prove that $CQ$ is parallel to $AP$. Prove also that $AC$ bisects $PQ$.

Q.

In a $\Delta ABC$ median AD is produced to X such that AD = DX. Prove that ABXC is a parallelogram.

Q. Question 1
Diagonals AC and BD of a parallelogram ABCD intersect each other at O. If OA = 3cm and OD = 2cm, determine the lengths of AC and BD.

Q.

Assertion : If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.
Reason : The opposite angles of a parallelogram are equal.
Which of the following is correct?

1. A is true and R is false.
2. A is false and R is true.
3. Both A and R are true and R is not the correct explanation of A.
4. Both A and R are true and R is the correct explanation of A.

Q.

M and N are points on opposite sides AD and BC of a parallelogram ABCD such that MN passes through the point of intersection O of its diagonals AC and BD. Show that MN is bisected at O.

Q.

A diagonal of a rectangle is inclined to one side of the rectangle at ${35}^{\circ }$. The acute angle between the diagonals is

(a) ${55}^{\circ }$

(b) ${70}^{\circ }$

(c) ${45}^{\circ }$

(d) ${50}^{\circ }$

Q. Question 5
E and F are points on diagonal AC of parallelogram ABCD such that AE = CF. Show that BFDE is a parallelogram.

Q.

Is quadrilateral ABCD a parallelogram?

I. Its opposite sides are equal.

II. Its opposite angles are equal.

The correct answer is: (a)/(b)/(c)/(d).

Assertion-and-Reason Type MCQ

Each question consists of two statements, namely, Assertion (A) and Reason (R). For selecting the correct answer, use the following code:

(a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).

(b) Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).

(c) Assertion (A) is true and Reason (R) is false.

(d) Assertion (A) is false and Reason (R) is true.

Q.

$\begin{array}{cc}\text{Assertion (A)}& \text{Reason (R)}\\ \text{The diagonals of a}||gm\text{bisect each}& \text{If the diagonals of a}||gm\text{are equal}\\ \text{other}& \text{and intersect at right angles then}\\ & \text{the parallelogram is a square.}\end{array}$

The correct answer is: (a)/(b)/(c)/(d).

Q. ABCD is a parallelogram with an area of $48c{m}^2$. If AC is a diagonal of the parallelogram, what is the area of $△ACD$ ?

1. $18c{m}^2$
2. $16c{m}^2$
3. $24c{m}^2$
4. $8c{m}^2$

Q. ABCD is a parallelogram as shown in figure. If $AB=2AD$ and P is the midpoint of AB, then $\angle CPD$ is equal to

1. $90°$
2. $60°$
3. $45°$
4. $135°$

Q.

ABCD is a parallelogram, if the two diagonals are equal, then by what criterion are the triangles ABD and ABC congruent

1. AAS

2. SAS

3. SSS

4. RHS

Q. If ABCD is a parallelogram where, AC and BD are the two diagonals. If BD is 6 cm more than AC and AE = 5 cm, find the length of BE.

1. 5 cm
2. 16 cm
3. 8 cm
4. 13 cm

Q.

A diagonal of a parallelogram divides it into :

1. None of the above

2. two congruent triangles

3. two equilateral triangles

4. two isosceles triangle

Q. In a parallelogram ABCD, if $AB=2x+5,CD=y+1,AD=y+5$ and $BC=3x-4,$ then the ratio of $AB:BC$ is equal to

1. $71:21$
2. $12:11$
3. $31:35$
4. $4:7$

Q.

Calculate the area of a triangle which is formed by a diagonal of a parallelogram having an area of 246 ${cm}^2$.

1. 123 ${cm}^2$

2. 132 ${cm}^2$

3. 231 ${cm}^2$

4. 321 ${cm}^2$

Q. Two adjacent sides of a parallelogram are 2i - 4j+ 5k and i - 2j - 3k. The unit vector parallel to its diagonal is?

Q. Question 3
Show that the diagonals of a parallelogram divide it into four triangles of equal area.

Q. In the following figure, ABCD is a parallelogram. If E is the mid-point of BC and AE is the bisector of ∠A, then prove that AB = $\frac{1}{2}$ AD.

Q. If a diagonal of a rectangle is inclined to one side of the rectangle at ${40}^{\circ }$, then the acute angle between the diagonals is _____.

1. $80°$
2. $70°$
3. $75°$
4. $60°$

Q.

This question consists of two statements, namely, Assertion (A) and Reason (R).For selecting the correct answer, use the following code:

(a) Both Assertion (A) and Reason (R) are true and Reason (R) correct explanation of Assertion (A).

(b) Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).

$\begin{array}{cc}Assertion\left(A\right)& Reason\left(R\right)\\ \text{The diagonals of a}||gmdivideit& \text{A diagonal of a}||gmdividesitinto\\ \text{into four triangles of equal area.}& \text{two diagonals of equal area.}\end{array}$

The correct answer is :(a) /(b) /(c) /(d)

Q. ABCD is a parallelogram with an area of $48c{m}^2$. If AC is a diagonal of the parallelogram, what is the area of $△ACD$ ?

1. $18c{m}^2$
2. $24c{m}^2$
3. $16c{m}^2$
4. $8c{m}^2$

Q.

In a parallelogram$ ABCD$, $ E$ and $ F$ are the mid-points of sides $ AB$ and $ CD$ respectively (see Fig.) . Show that the line segments $ AF$ and $ EC$ trisect the diagonal $ BD$.

Q.

Which of the following statements are true (T) and which are false (F)?

In a parallelogram, the diagonals bisect each other

Q.

ABCD is a parallelogram whose diagonals AC and BD intersect at O.A line through O intersects AB at P and DC at Q. Prove that $ar\left(△POA\right)=ar\left(△QOC\right).$