Q1. ABCD is a parallelogram with base AB. E and F are the midpoints of sides BC and CD of
the parallelogram. Prove that the area of triangle AEF is equal to 3/8th of the area of the
parallelogram.
Q2. P and Q are any the mid-points of the sides DC and AD respectively of a parallelogram
ABCD. Prove that area of triangles APB and BQC are equal.
Q3. A parallelogram has base 10 cm and height 6 cm. Find the area of a triangle on the
same base and of same height.
Q4. A parallelogram ABCD has E, F, G and H as the points on the sides AB, BC, CD and AD
respectively. If area of EFGH is half of the area of triangle ABC, then prove that E, F, G and
H are the mid-points of the respective sides.
Q5. In a triangle ABC, D is a point on side BC and E is a point on AD such that area of
triangle ABE = area of triangle ACE. Show that AD is median of triangle ABC.
Q6. In a parallelogram ABCD with base 8 cm and height 4.5 cm, diagonals AD and BC
intersect at O. Find the area of triangle ABO.
Q7. If the area of a parallelogram is 125 square units and height is equal to 1/5 of base,
then:
(a) Find the height of the parallelogram.
(b) Find the area of triangle with same base and height as parallelogram.
Q8. Prove that diagonals of a rhombus divide it into four equal triangles.
Q9. ABCD is a quadrilateral in which DC is produced to point E such that BE || AC. Prove
that ar(∆ADE) = ar(quadrilateral ABCD).
Q10. In a parallelogram PQRS, PM
6 cm, find:
RS and SN
(a)
length of QR.
(b)
Area of parallelogram PQRS.
(c)
Area of ∆PQR.
QR. If PQ = 8 cm, PM = 5 cm and SN =
Q11. PQRS is a trapezium with PQ || SR. A line parallel to PQ intersects PQ at X and QR at
Y. Prove that ar (PSX) = ar (PRY).
Q12. In a ∆ABC, M is mid-point of BC and N is a point on AB such that AN : NB = 2 : 1. If
area of ∆AMN is 3.5 cm2 , find the area of ∆ABC.
Q13. Prove that lines joining the mid-points of adjacent sides of a parallelogram also form
a parallelogram and its area is half of original parallelogram.
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Q14. Diagonals PR and QS of a quadrilateral PQRS intersect at O in such a way that ar
(POS) = ar (QOR). Prove that PQRS is a trapezium.
Q15. In ∆ABC, D and E are two points on BC such that BD=DE=EC. Prove that: ar (ADE) =
1
ar (ABC).
3
Q16. In the figure shown, if ar (ADE) = ar (BCF) then prove that ABCD, DCFE and ABFE
are all parallelograms.
Q17. ABC is an equilateral triangle such that D is the mid-point of BC. BDE is also an
equilateral triangle on the other side of ABC. Prove that ar(ABC) = 4 ar(BDE).
Q18. Show that a median of a triangle divides it into two triangles of equal areas.
Q19. In ∆ABC, D and E are mid-points of AB and AC respectively. Find the ratio of
ar(∆ABC) : ar(∆ADE).
Q20. The diagonals AC and BD of a quadrilateral ABCD intersect at O such that OB = OD.
If AB = CD, then show that ABCD is a parallelogram.
Answers:
A1. Hint: Join B and D and use area properties of parallelogram and triangle.
A2. Hint: Use properties of parallelogram and triangle
A3. 30 cm2
A4. Hint: Use properties of parallelogram and triangle
A5. Hint: Use properties of triangle
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A6. 9 cm2
A7. (a) 5 units
(b) 62.5 cm2
A8. Hint: Use properties of parallelogram
A9. Hint: Use area property of quadrilateral
A10. (a)
20
cm
3
(b) 40 cm2
(c) 20 cm2
A11. Hint: Join RX.
A12. 10.5 cm2
A13. Hint: Use properties of parallelogram
A14. Hint: Show that properties of trapezium are satisfied.
A15. Hint: Use properties of triangle
A16. Hint: Use properties of parallelogram
A17. Hint: Show that BE || AC and DE || AB
A18. Hint: Use properties of triangle
A19. 4: 1
A20. Hint: Show that properties of parallelogram are satisfied.
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