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Areas of Parallelograms and Triangles CBSE Class 9

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Question 1/20

Q 1. ABCD is a parallelogram. If AB = 15 cm, AE = 14 cm, CF = 7 cm, then AD =   

Solution

From the figure,

Area of a parallelogram ABCD with base DC and

AE height = DC × AE

Area of a parallelogram ABCD with base AD and

height CF = AD × CF

DC × AE = AD × CF

  

Q 2. If ar(parallelogram ABCD) = 25 cm2 and is on the same base CD, then ar(BCD) =

Solution

If ar(ABCD) = 25 cm2 and is on the same base CD, then ar(BCD) = 25/2 = 12.5 cm2.

Q 3. Which of the following is incorrect? 

Solution

Two congruent figures have equal areas, but the converse is not true. 

Q 4. ABCD is a rectangle in which CD = 6 cm, AD = 8 cm. Find the area of parallelogram CDEF.   

Solution

ABCD is a rectangle in which CD = 6 cm, AD = 8 cm. The area of parallelogram CDEF is 6 × 8 = 48 cm2. 

Q 5. If each diagonal of a quadrilateral separates it into two triangles of equal areas, then the quadrilateral is a 

Solution

If each diagonal of a quadrilateral separates it into two triangles of equal areas, then the quadrilateral is a parallelogram. 

Q 6. If a triangle and a parallelogram are on the same base and between the same parallels, then the ratio of the area of the triangle to the area of the parallelogram is 

Solution

If a triangle and a parallelogram are on the same base and between the same parallels, then the ratio of the area of the triangle to the area of the parallelogram is 1:2. 

Q 7. In a ABC, D, E and F are the mid-points of sides BC, CA and AB, respectively. If ar(ABC) = 30 cm2, then ar(quadrilateral FDCE) = 

Solution

  

When the mid-points are joined, all triangles are equal.

Area of any of the triangles in ABC = 30/4 = 7.5 cm2 ar(quadrilateral FDCE) = 2 × 7.5 = 15 cm2

Q 8. ABCD is a parallelogram. P is any point on CD. If ar(DPA) = 30 cm2 and ar(APC) = 10 cm2, then ar(ADC) = 

Solution

  

ar(ADC) = ar(DPA) + ar(APC) = 40 cm2

Q 9. In the given figure, EBCF is a parallelogram and D is the mid-point of BC. If ar(ADC) = 20 cm2, find the area of parallelogram EBCF.   

Solution

ar(EBCF) = 4 × ar(ADC) = 80 cm2 

Q 10. In the given figure, EBCF is a parallelogram and D is the mid-point of BC. If ar(ADC) = 12 cm2, then find the area of parallelogram EBCF.   

Solution

ar(EBCF) = 4 × ar(ADC) = 48 cm2 

Q 11. DCFE is a parallelogram. ABCD is a rectangle in which CD = 6 cm, AD = 8 cm. Find the area of GEF.   

Solution

ABCD is a rectangle in which CD = 6 cm, AD = 8 cm. The area of parallelogram CDEF is 6 × 8 = 48 cm2.

ar(GEF) = 48/2 = 24 cm2

Q 12. If ABCD is a parallelogram, then ar(AFB) is   

Solution

  

Q 13. In ABC, D, E and F are the mid-points of sides BC, CA and AB, respectively. If ar(ABC) = 20 cm2, then ar(trapezium FBCE) = 

Solution

  

Area of any of the triangles in ABC = 20/4 = 5 cm2.

ar(trapezium FBCE) = 3 × 5 = 15 cm2.

Q 14. In ABC, D, E and F are the mid-points of sides BC, CA and AB, respectively. If ar(ABC) = 16 cm2, then ar(trapezium FBCE) = 

Solution

  

Area of any of the triangles in ABC = 16/4 = 4 cm2.ar(trapezium FBCE) = 3 × 4 = 12 cm2.

Q 15. Diagonals AC and BD of trapezium ABCD, in which AB is parallel to DC, intersect each other at O. The triangle which is equal in area to AOD is 

Solution

Diagonals AC and BD of trapezium ABCD, in which AB is parallel to DC, intersect each other at O. The triangle which is equal in area to AOD is BOC.

  

As AOD BOC by the ASA criteria.

Q 16. A, B, C and D are the mid-points of the sides of a parallelogram PQRS. If ar(PQRS) = 50 cm2, then ar(ABCD) = 

Solution

  

If P, Q, R, S are the mid-points of the sides AB, BC, CD and DA of a quadrilateral ABCD. Show that PQRS is a parallelogram such that ar(PQRS) = 1/2 ar(ABCD). 

Q 17. If the diagonals AC and BD of a quadrilateral ABCD intersect at O and separate the quadrilateral into four triangles of equal area, then the quadrilateral ABCD is a 

Solution

If the diagonals AC and BD of a quadrilateral ABCD intersect at O and separate the quadrilateral into four triangles of equal area, then the quadrilateral ABCD is a parallelogram. 

Q 18. If CL is perpendicular to AB and DM is perpendicular to AB, where ar(ACB) = ar(ADB), then   

Solution

Triangles are on the same base and between the same parallels. 

Q 19. If a parallelogram and a rectangle are on the same base and between the same parallels, then the perimeter of a parallelogram is _____ than the perimeter of a rectangle. 

Solution

If a parallelogram and a rectangle are on the same base and between the same parallels, then the perimeter of a parallelogram is greater than the perimeter of a rectangle. 

Q 20. ABCD and ABFE are parallelograms and ar(EABC) = 17 cm2 and ar(ABCD) = 25 cm2, then ar(BCF) =   

Solution

Triangles on the same base and between the same parallels are equal in area.

ar(BCF) = ar(ADE)

ar(BCF) = ar(ABCD) - ar(EABC)

ar(BCF) = 25 - 17 = 8 cm2

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