Properties of Parallelograms

Q. In the given parallelogram ABCD, E and F are the mid-points of sides AB and CD, respectively. Diagonal DB intersects AF and EC at P and Q, respectively. If PB = 2cm, find the length of BD.

1. 4 cm
2. 3 cm
3. 6 cm
4. Insufficient data

Q. ABCD is a quadrilateral and BD is one of its diagonals as shown in fig. Show that ABCD is a parallelogram and find its area.

Q. In the figure, the diagonals of a quadrilateral split it into four triangles. The areas of three of them are shown in the picture:

Calculate the area of the whole quadrilateral.

Q. 49. construct a traingle pqr if pq is equal to 7.5 centimetre , angle q is equal to 90 degree and pr _rq is epual to 4 centimetre

Q. 21 D, E and F are respectively the mid points of sides AB, BC & CA, respectively of a triangle ABC. Prove that by joining these mid points D, E & F, the triangle ABC is divided into 4 congruent triangles.

Q. D and E are points on the sides AB and AC respectively of △ ABC such that DE\parallel BC and divides△ ABC in two parts , equal in area.Find BD/AB

Q. ABCD is a parallelogram in which }DC=14cm, AC=16cm and area of parallelogram is }80\sqrt3cm^2 . } The length of }AD can be equal to } (1) }18cm(2) 10cm(3) }13.5cm(4)12cm .

Q.

Prove that if both pairs of opposite angles of a quadrilateral are equal, then it is a parallelogram. [3 MARKS]

Q.

If AD is median of ΔABC and P is a point on AC such that area (ΔADP) : area (ΔABD) = 2 : 3, then ar(ΔPDC) : ar(ΔABC) is

1. 1 : 5

2. 1 : 6

3. 2 : 5

4. 3 : 5

Q.

In the given figure, ABCP is a parallelogram and P is the mid point of CD. Then find which of the following option is correct?

1. $OA=\frac{1}{2}CD$

2. $PD=AP$

3. $AP=\frac{1}{2}BC$

4. $OB=\frac{1}{2}BD$

Q. ABC is a triangle in which DE\vert\vert BCsuch that AD/DB=3/2.Prove that △ DEF∼△ CBF and also find the ratio ar(△ DEF):ar(△ DEC), where F is the point of intersection of DC and BE

Q. ABCD is a parelle\log ram, AB is produced to E such that AB=BE. Prove that ED bisects BC.

Q. In the following figure, ABC and BDE are two equilateral triangles such that D is the mid-point of BC. If AE intersects BC at F,
Show that $ar\left(BDE\right)=\frac{1}{2}ar\left(BAE\right)$

Q.

Given, BC = 5 cm and the corresponding altitude AP = 8 cm. Find the area of the parallelogram ABCD.

[2 Marks]

Q.

Points A(0, 0), B(4, 0), C(6, 6) and D(2, 6) are joined to form quadrilateral ABCD. Point C(6, 6) and point B(4, 0) are joined to point F(8, 0). The quadrilateral DBFC is now completed. Which of the following is correct?

1. Area of quad ABCD = 1 × Area of quad DBFC

2. Area of quad ABCD = (1/4) × Area of quad DBFC

3. Area of quad ABCD = (1/2) × Area of quad DBFC

4. Area of quad ABCD = 2 × Area of quad DBFC

Q. In the following figure, ABC and BDE are two equilateral triangles such that D is the mid-point of BC. If AE intersects BC at F,
Show that $ar\left(BDE\right)=\frac{1}{2}ar\left(BAE\right)$ [4 MARKS]

Q.

In the parallelogram, D is the mid point of side PQ. Which of the following options is NOT true?

1. PQ = RS

2. PD = $\frac{1}{2}$ RS

3. DQ = $\frac{1}{2}$RS

4. PQ = 3PD

Q.

The area of triangle with vertices $A(x_1,y_1),B(x_2,y_2)\text{and}C(x_3,y_3)$ is:

1. $\frac{1}{2}\left[x_1\right(y_2-y_3)+x_3(y_3-y_1)+x_1(y_1-y_2\left)\right]$

2. All of the above.

3. $\frac{1}{2}\left[x_2\right(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2\left)\right]$

4. $\frac{1}{2}\left[x_1\right(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2\left)\right]$

Q. If E is the mid point of median AD in triangle ABC, then prove that :
Area ($\Delta ABE$) = $\frac{1}{4}$ Area ($\Delta ABC$).

Q.

In a triangle ABC, E is the mid-point of median AD. Show that ar (ΔBED) = $\frac{1}{4}$ar(ΔABC) [2 MARKS]

Q. In a ||gm ABCD, the bisector of angle A also bisects BC at X. Find AB : AD 1. 2 : 1 2. 1 : 2 3. 3 : 2 4. 2 : 3

Q. In an equilateral triangle ABC a circle passes through vertex B such that it touches the midpoint of AC and intersect the sides AB and BC at M and N respectively then which of the following is not correct (1)TC=2AM (2)MB=3CN (3)AM=CN (4)BN=AT

Q.

ABC is a triangle in which E is mid-point of median AD. Which of the following is not true?

1. area of $\Delta BED=\frac{1}{4}$ area of $\Delta ABC$

2. area of $\Delta BED$ = area of $\Delta BAE$

3. area of $\Delta ABD$ = area of $\Delta ADC$

4. area of $\Delta BED=\frac{1}{2}$ area of $\Delta ABC$

Q. Prove that the line segments joining the mid-points of the adjacent sides of a quadrilateral, taken in order form a parallelogram.

Q. The area of a triangle is $\frac{1}{2}\times base\times height$

1. True
2. False

Q. Question 5 (i)
D, E and F are respectively the mid-points of the sides BC, CA and AB of triangle ABC Show that BDEF is a parallelogram.

Q. СP QВ30. The base BC of an isosceles triangle ABC is produced to D and a straline DQ is drawn cutting AC at P and meeting AB in Q. Prove that AP isgreater than AQ

Q. 27. if each side of parallelogram is increased by 50% then the overall increase in the area of parallelogram will equal to

Q.

State which statement is/are true of the following figure.

1. $\frac{1}{2}$ $\times$ area of AFEB = area of $\Delta$ DEF

2. area of $\Delta$ ABE = area of $\Delta$ DEF

3. area of $\Delta$ AED = $\frac{1}{2}$ $\times$ area of ABCD

4. All of the above

Q. In a triangle $ABC,E$ is the mid-point of median $AD$. Show that $area\left(BED\right)=\frac{1}{4}area\left(ABC\right)$.