# Area and Midpoint Theorems

- Parallelograms on the same base and between the same parallels are equal in area.
- The area of a parallelogram is equal to the area of a rectangle on the same base and between the same parallels.
- Area of a triangle is half that of a parallelogram on the same base and between the same parallels.
- Triangles on the same base and between the same parallels are equal in area.
- The line segment joining the mid-points of any two sides of a triangle is parallel to the third side and is equal to half of it.
- The line drawn through the mid-point of one side of a triangle parallel to another side bisects the third side.

## Exercise

- In the following diagram, ABCD is a parallelogram. P and Q are points on the sides
DC and BC respectively. Prove that

(i) area of APB = area of AQD

(ii) area of APB = area of ABQ +area of DQC.

- In the following figure, area of the parallelogram ABCD is 29 cm². If AB = 5.8 cm, find the height of the parallelogram EFCD.
- In the following figure, AB || DC and area of ABD is 24 sq.
units. If AB = 8 units, find the height of ABC.

- In the following figure, AB || DC || EF, DA || EB and DE || AF. Prove that area of || gm DEFH = area of || gm ABCD.

[**Hint.**Area of || gm ABCD = area of || ADEG = area of || gm DEFH.]

- In the following figure, DE is parallel to the side BC of ABC. BE and CD intersect at O. Prove that

(i) area of BED = area of CED.

(ii) area of BOD = area of COE.

(iii) area of ABE = area of ADC.

- In ABC, P and Q are mid-points of the sides AB and AC
respectively. If BC = 6 cm, AB = 5.4 cm and AC = 5 cm, calculate the perimeter of the
quadrilateral PBCQ.

- ABCD is a rectangle. E, F, G and H are the mid-points of the
sides AB, BC, CD and DA respectively. Prove that EFGH is a rhombus.

[**Hint.**EF is parallel and half of AC and HG is parallel and half of AC

=> EFGH is a || gm.

Also FG = BD/2 but BD = AC => FG = HG.]

- In ABC, D, E and F are the mid-points of the sides AB, BC and CA respectively. Prove that the
S ADF, DBE, ECF and EFD are congruent to each other.

- In ABC, E and F are the mid-points of the sides BC and CA
respectively. EF = 4 cm and ED || CA. If area of ||gm DEFA = 16 cm2, calculate.

(i) AB (ii) area of BED

(iii) area of trapezium DECA.

## Answers

**2.**5 cm

**3.**6 units

**6.**14.2 cm

**9.**(i) 8 cm (ii) 8 cm² (iii) 24 cm²