# Properties of Diagonal of a Parallelogram

## Trending Questions

**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∘ and ∠BDC=50∘ then ∠OAB=

40∘

50∘

10∘

90∘

**Q.**

The diagonals AC and BD of a parallelogram ABCD intersect each other at the point O such that ∠DAC=30∘ and ∠AOB=70∘. Then, ∠DBC= ?

(a) 40∘

(b) 35∘

(c) 45∘

(d) 50∘

**Q.**

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

(a) 45∘

(b) 60∘

(c) 90∘

(d) 30∘

**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 ΔABC 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?

- A is true and R is false.
- A is false and R is true.
- Both A and R are true and R is not the correct explanation of A.
- 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∘. The acute angle between the diagonals is

(a) 55∘

(b) 70∘

(c) 45∘

(d) 50∘

**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.**

Assertion (A)Reason (R)The diagonals of a ||gm bisect eachIf the diagonals of a ||gm are equalotherand intersect at right angles then the parallelogram is a square.

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

**Q.**ABCD is a parallelogram with an area of 48 cm2. If AC is a diagonal of the parallelogram, what is the area of △ACD ?

- 18 cm2
- 16 cm2
- 24 cm2
- 8 cm2

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

- 90°
- 60°
- 45°
- 135°

**Q.**

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

AAS

SAS

SSS

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.

- 5 cm
- 16 cm
- 8 cm
- 13 cm

**Q.**

A diagonal of a parallelogram divides it into :

- None of the above
two congruent triangles

two equilateral triangles

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

- 71:21
- 12:11
- 31:35
- 4:7

**Q.**

Calculate the area of a triangle which is formed by a diagonal of a parallelogram having an area of 246 cm2.

123 cm2

132 cm2

231 cm2

321 cm2

**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∘, then the acute angle between the diagonals is _____.

- 80°
- 70°
- 75°
- 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).

Assertion (A)Reason (R)The diagonals of a ||gm divide itA diagonal of a ||gm divides it intointo four triangles of equal area.two diagonals of equal area.

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

**Q.**ABCD is a parallelogram with an area of 48 cm2. If AC is a diagonal of the parallelogram, what is the area of △ACD ?

- 18 cm2
- 24 cm2
- 16 cm2
- 8 cm2

**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(△POA)=ar(△QOC).