RD Sharma MCQs: Areas of Parallelograms and Triangles | Mathematics (Maths) Class 9 PDF Download

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Question:46
Two parallelograms are on the same base and between the same parallels. The ratio of their areas is
a
1 : 2
b
2 : 1
c
1 : 1
d
3 : 1
Solution:
Given: Two parallelogram with the same base and between the same parallels.
To find: Ratio of their area of two parallelogram with the same base and between the same parallels.
Page 2


                     
  
 
  
  
  
 
  
 
        
  
     
 
  
  
  
  
 
  
   
Question:46
Two parallelograms are on the same base and between the same parallels. The ratio of their areas is
a
1 : 2
b
2 : 1
c
1 : 1
d
3 : 1
Solution:
Given: Two parallelogram with the same base and between the same parallels.
To find: Ratio of their area of two parallelogram with the same base and between the same parallels.
Calculation: We know that “Two parallelogram with the same base and between the same parallels, are equal in area”
Hence their ratio is ,
So the correct answer is ,i.e. option c
.
 
Question:47
A triangle and a parallelogram are on the same base and between the same parallels. The ratio of the areas of triangle
and parallelogram is
a
1 : 1
b
1 : 2
c
2 : 1
d
1 : 3
Solution:
Given: A triangle and a parallelogram with the same base and between the same parallels.
To find: The ratio of the area of a triangle and a parallelogram with the same base and between the same parallels.
Calculation: We know that,” the area of a triangle is half the area of a parallelogram with the same base and between the
same parallels.”
Hence the ratio of the area of a triangle and a parallelogram with the same base and between the same parallels is 
Therefore the correct answer is option is b
.
 
Question:48
Let ABC be a triangle of area 24 sq. units and PQR be the triangle formed by the mid-points of the sides of ? ABC. Then
the area of ?PQR is
a
12 sq. units
b
6 sq. units
c
4 sq. units
d
3 sq. units
Solution:
Given: 1
The Area of ?ABC = 24 sq units.
2
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Question:46
Two parallelograms are on the same base and between the same parallels. The ratio of their areas is
a
1 : 2
b
2 : 1
c
1 : 1
d
3 : 1
Solution:
Given: Two parallelogram with the same base and between the same parallels.
To find: Ratio of their area of two parallelogram with the same base and between the same parallels.
Calculation: We know that “Two parallelogram with the same base and between the same parallels, are equal in area”
Hence their ratio is ,
So the correct answer is ,i.e. option c
.
 
Question:47
A triangle and a parallelogram are on the same base and between the same parallels. The ratio of the areas of triangle
and parallelogram is
a
1 : 1
b
1 : 2
c
2 : 1
d
1 : 3
Solution:
Given: A triangle and a parallelogram with the same base and between the same parallels.
To find: The ratio of the area of a triangle and a parallelogram with the same base and between the same parallels.
Calculation: We know that,” the area of a triangle is half the area of a parallelogram with the same base and between the
same parallels.”
Hence the ratio of the area of a triangle and a parallelogram with the same base and between the same parallels is 
Therefore the correct answer is option is b
.
 
Question:48
Let ABC be a triangle of area 24 sq. units and PQR be the triangle formed by the mid-points of the sides of ? ABC. Then
the area of ?PQR is
a
12 sq. units
b
6 sq. units
c
4 sq. units
d
3 sq. units
Solution:
Given: 1
The Area of ?ABC = 24 sq units.
2
?PQR is formed by joining the midpoints of ?ABC
To find: The area of ?PQR
Calculation: In ?ABC, we have
Since Q and R are the midpoints of BC and AC respectively.
 PQ || BA  PQ || BP
Similarly, RQ || BP. So BQRP is a parallelogram.
Similarly APRQ and PQCR are parallelograms.
We know that diagonal of a parallelogram bisect the parallelogram into two triangles of equal area.
Now, PR is a diagonal of APQR.
? Area of ?APR = Area of ?PQR ……1
Similarly,
PQ is a diagonal of  PBQR
? Area of ?PQR = Area of ?PBQ ……2
QR is the diagonal of  PQCR
? Area of ?PQR = Area of ?RCQ ……3
From 1
, 2
, 3
we have
Area of ?APR = Area of ?PQR = Area of ?PBQ = Area of ?RCQ
But
Area of ?APR + Area of ?PQR + Area of ?PBQ + Area of ?RCQ = Area of ?ABC
4(Area of ?PBQ) = Area of ?ABC
? Area of ?PBQ 
Hence the correct answer is option b
.
Question:49
The median of a triangle divides it into two
a
congruent triangle
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Question:46
Two parallelograms are on the same base and between the same parallels. The ratio of their areas is
a
1 : 2
b
2 : 1
c
1 : 1
d
3 : 1
Solution:
Given: Two parallelogram with the same base and between the same parallels.
To find: Ratio of their area of two parallelogram with the same base and between the same parallels.
Calculation: We know that “Two parallelogram with the same base and between the same parallels, are equal in area”
Hence their ratio is ,
So the correct answer is ,i.e. option c
.
 
Question:47
A triangle and a parallelogram are on the same base and between the same parallels. The ratio of the areas of triangle
and parallelogram is
a
1 : 1
b
1 : 2
c
2 : 1
d
1 : 3
Solution:
Given: A triangle and a parallelogram with the same base and between the same parallels.
To find: The ratio of the area of a triangle and a parallelogram with the same base and between the same parallels.
Calculation: We know that,” the area of a triangle is half the area of a parallelogram with the same base and between the
same parallels.”
Hence the ratio of the area of a triangle and a parallelogram with the same base and between the same parallels is 
Therefore the correct answer is option is b
.
 
Question:48
Let ABC be a triangle of area 24 sq. units and PQR be the triangle formed by the mid-points of the sides of ? ABC. Then
the area of ?PQR is
a
12 sq. units
b
6 sq. units
c
4 sq. units
d
3 sq. units
Solution:
Given: 1
The Area of ?ABC = 24 sq units.
2
?PQR is formed by joining the midpoints of ?ABC
To find: The area of ?PQR
Calculation: In ?ABC, we have
Since Q and R are the midpoints of BC and AC respectively.
 PQ || BA  PQ || BP
Similarly, RQ || BP. So BQRP is a parallelogram.
Similarly APRQ and PQCR are parallelograms.
We know that diagonal of a parallelogram bisect the parallelogram into two triangles of equal area.
Now, PR is a diagonal of APQR.
? Area of ?APR = Area of ?PQR ……1
Similarly,
PQ is a diagonal of  PBQR
? Area of ?PQR = Area of ?PBQ ……2
QR is the diagonal of  PQCR
? Area of ?PQR = Area of ?RCQ ……3
From 1
, 2
, 3
we have
Area of ?APR = Area of ?PQR = Area of ?PBQ = Area of ?RCQ
But
Area of ?APR + Area of ?PQR + Area of ?PBQ + Area of ?RCQ = Area of ?ABC
4(Area of ?PBQ) = Area of ?ABC
? Area of ?PBQ 
Hence the correct answer is option b
.
Question:49
The median of a triangle divides it into two
a
congruent triangle
b
isosceles triangles
c
right triangles
d
triangles of equal areas
Solution:
Given: A triangle with a median.
Calculation: We know that a ,”median of a triangle divides it into two triangles of equal area.”
Hence the correct answer is option d
.
 
Question:50
In a ?ABC, D, E, F are the mid-points of sides BC, CA and AB respectively. If ar (?ABC) = 16cm
2
, then ar (trapezium
FBCE) =
a
4 cm
2
b
8 cm
2
c
12 cm
2
d
10 cm
2
Solution:
Given: In ?ABC
1
D is the midpoint of BC
2
E is the midpoint of CA
3
F is the midpoint of AB
4
Area of ?ABC = 16 cm
2
To find: The area of Trapezium FBCE
Calculation: Here we can see that in the given figure,
Area of trapezium FBCE = Area of ||
gm
 FBDE + Area of ?CDE
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Question:46
Two parallelograms are on the same base and between the same parallels. The ratio of their areas is
a
1 : 2
b
2 : 1
c
1 : 1
d
3 : 1
Solution:
Given: Two parallelogram with the same base and between the same parallels.
To find: Ratio of their area of two parallelogram with the same base and between the same parallels.
Calculation: We know that “Two parallelogram with the same base and between the same parallels, are equal in area”
Hence their ratio is ,
So the correct answer is ,i.e. option c
.
 
Question:47
A triangle and a parallelogram are on the same base and between the same parallels. The ratio of the areas of triangle
and parallelogram is
a
1 : 1
b
1 : 2
c
2 : 1
d
1 : 3
Solution:
Given: A triangle and a parallelogram with the same base and between the same parallels.
To find: The ratio of the area of a triangle and a parallelogram with the same base and between the same parallels.
Calculation: We know that,” the area of a triangle is half the area of a parallelogram with the same base and between the
same parallels.”
Hence the ratio of the area of a triangle and a parallelogram with the same base and between the same parallels is 
Therefore the correct answer is option is b
.
 
Question:48
Let ABC be a triangle of area 24 sq. units and PQR be the triangle formed by the mid-points of the sides of ? ABC. Then
the area of ?PQR is
a
12 sq. units
b
6 sq. units
c
4 sq. units
d
3 sq. units
Solution:
Given: 1
The Area of ?ABC = 24 sq units.
2
?PQR is formed by joining the midpoints of ?ABC
To find: The area of ?PQR
Calculation: In ?ABC, we have
Since Q and R are the midpoints of BC and AC respectively.
 PQ || BA  PQ || BP
Similarly, RQ || BP. So BQRP is a parallelogram.
Similarly APRQ and PQCR are parallelograms.
We know that diagonal of a parallelogram bisect the parallelogram into two triangles of equal area.
Now, PR is a diagonal of APQR.
? Area of ?APR = Area of ?PQR ……1
Similarly,
PQ is a diagonal of  PBQR
? Area of ?PQR = Area of ?PBQ ……2
QR is the diagonal of  PQCR
? Area of ?PQR = Area of ?RCQ ……3
From 1
, 2
, 3
we have
Area of ?APR = Area of ?PQR = Area of ?PBQ = Area of ?RCQ
But
Area of ?APR + Area of ?PQR + Area of ?PBQ + Area of ?RCQ = Area of ?ABC
4(Area of ?PBQ) = Area of ?ABC
? Area of ?PBQ 
Hence the correct answer is option b
.
Question:49
The median of a triangle divides it into two
a
congruent triangle
b
isosceles triangles
c
right triangles
d
triangles of equal areas
Solution:
Given: A triangle with a median.
Calculation: We know that a ,”median of a triangle divides it into two triangles of equal area.”
Hence the correct answer is option d
.
 
Question:50
In a ?ABC, D, E, F are the mid-points of sides BC, CA and AB respectively. If ar (?ABC) = 16cm
2
, then ar (trapezium
FBCE) =
a
4 cm
2
b
8 cm
2
c
12 cm
2
d
10 cm
2
Solution:
Given: In ?ABC
1
D is the midpoint of BC
2
E is the midpoint of CA
3
F is the midpoint of AB
4
Area of ?ABC = 16 cm
2
To find: The area of Trapezium FBCE
Calculation: Here we can see that in the given figure,
Area of trapezium FBCE = Area of ||
gm
 FBDE + Area of ?CDE
Since D and E are the midpoints of BC and AC respectively.
? DE || BA  DE || BF
Similarly, FE || BD. So BDEF is a parallelogram.
Now, DF is a diagonal of ||
gm
 BDEF.
? Area of ?BDF = Area of ?DEF ……1
Similarly,
DE is a diagonal of ||
gm
 DCEF
? Area of ?DCE = Area of ?DEF ……2
FE is the diagonal of ||
gm
 AFDE
? Area of ?AFE = Area of ?DEF ……3
From 1
, 2
, 3
we have
Area of ?BDF = Area of ?DCF = Area of ?AFE = Area of ?DEF
But
Area of ?BDF + Area of ?DCE + Area of ?AFE + Area of ?DEF = Area of ?ABC
? 4 Area of ?BDF = Area of ?ABC
Area of ?BDF = Area of ?DCE = Area of ?AFE = Area of ?DEF = 4 cm
2
 …….4
Now
Hence we get
Area of trapezium FBCE 
There fore the correct answer is option c
.
Question:51
ABCD is a parallelogram. P is any point on CD. If ar (?DPA) = 15 cm2 and ar (?APC) = 20 cm
2
, then ar (?APB) =