RD Sharma Solutions: Areas of Parallelograms and Triangles- 2

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Question:2
If the given figure, ABCD is a parallelogram, AE ? DC and CF ? AD. If AB = 16 cm. AE = 8 cm and CF = 10 cm, find AD.
 
Solution:
Given: Here in the question it is given
1 ABCD is a parallelogram,
2 and
3 , AB = 16 cm
4 AE = 8cm
5 CF = 10cm
To Find : AD =?
Calculation: We know that formula for calculating the
Page 2


                  
 
                
    
 
 
Question:2
If the given figure, ABCD is a parallelogram, AE ? DC and CF ? AD. If AB = 16 cm. AE = 8 cm and CF = 10 cm, find AD.
 
Solution:
Given: Here in the question it is given
1 ABCD is a parallelogram,
2 and
3 , AB = 16 cm
4 AE = 8cm
5 CF = 10cm
To Find : AD =?
Calculation: We know that formula for calculating the
Therefore,
Area of paralleogram ABCD = DC × AE
TakingbaseasDCandHeightasAE
Area of paralleogram ABCD = AB × AE
AB = DCasoppositesideoftheparallelogramareequal
Therefore,
Area of paralleogram ABCD = 16 × 8 ……
1
Taking the base of Parallelogram ABCD as AD we get
Area of paralleogram ABCD = AD × CF
takingbaseasADandheightasCF
Area of paralleogram ABCD = AD × 10 ……
2
Since equation 1 and 2 both represent the Area of the same Parallelogram ABCD , both should be equal.
Hence fro equation
1 and
2,
This means that,
Hence we get the result as
Question:3
In Q.No. 1, if AD = 6 cm, CF = 10 cm, and AE = 8 cm, find AB.
Solution:
Given: Here in the question it is given that
1 ABCD is a parallelogram,
2 and
3 
4 AD = 6 cm
5 AE = 8cm
6 CF = 10cm
To Find : AB =?
Calculation: We know that formula for calculating the
Area of paralleogram = base × height
Page 3


                  
 
                
    
 
 
Question:2
If the given figure, ABCD is a parallelogram, AE ? DC and CF ? AD. If AB = 16 cm. AE = 8 cm and CF = 10 cm, find AD.
 
Solution:
Given: Here in the question it is given
1 ABCD is a parallelogram,
2 and
3 , AB = 16 cm
4 AE = 8cm
5 CF = 10cm
To Find : AD =?
Calculation: We know that formula for calculating the
Therefore,
Area of paralleogram ABCD = DC × AE
TakingbaseasDCandHeightasAE
Area of paralleogram ABCD = AB × AE
AB = DCasoppositesideoftheparallelogramareequal
Therefore,
Area of paralleogram ABCD = 16 × 8 ……
1
Taking the base of Parallelogram ABCD as AD we get
Area of paralleogram ABCD = AD × CF
takingbaseasADandheightasCF
Area of paralleogram ABCD = AD × 10 ……
2
Since equation 1 and 2 both represent the Area of the same Parallelogram ABCD , both should be equal.
Hence fro equation
1 and
2,
This means that,
Hence we get the result as
Question:3
In Q.No. 1, if AD = 6 cm, CF = 10 cm, and AE = 8 cm, find AB.
Solution:
Given: Here in the question it is given that
1 ABCD is a parallelogram,
2 and
3 
4 AD = 6 cm
5 AE = 8cm
6 CF = 10cm
To Find : AB =?
Calculation: We know that formula for calculating the
Area of paralleogram = base × height
Therefore,
Area of paralleogram ABCD = DC × AE
TakingbaseasDCandHeightasAE
Area of paralleogram ABCD = AB × AE
AB = DCasoppositesideoftheparallelogramareequal
Therefore, Area of paralleogram ABCd = 16 × 8
Area of Parallelogram ABCD = AB× 8 ……
1
Taking the base of Parallelogram ABCD as AD we get
Area of paralleogram ABCD = AD × CF
takingbaseasADandheightasCF
Area of paralleogram ABCD = 6 × 10 ……
2
Since equation 1and 2 both represent the Area of the same Parallelogram ABCD , both should be equal.
Hence equation 1 is equal to equation 2
Which means that,
Hence we got the measure of AB equal to 
 
Question:4
Let ABCD be a parallelogram of area 124 cm
2
. If E and F are the mid-points of sides AB and CD respectively, then find
the area of parallelogram AEFD.
Solution:
Given: Here in the question it is given that
1 Area of paralleogram ABCD = 124 cm
2
2 E is the midpoint of AB, which means 
3 F is the midpoint of CD, which means 
To Find : Area of Parallelogram AEFD
Calculation: We know that formula for calculating the
Area of Parallelogram = base × height
Therefore,
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Question:2
If the given figure, ABCD is a parallelogram, AE ? DC and CF ? AD. If AB = 16 cm. AE = 8 cm and CF = 10 cm, find AD.
 
Solution:
Given: Here in the question it is given
1 ABCD is a parallelogram,
2 and
3 , AB = 16 cm
4 AE = 8cm
5 CF = 10cm
To Find : AD =?
Calculation: We know that formula for calculating the
Therefore,
Area of paralleogram ABCD = DC × AE
TakingbaseasDCandHeightasAE
Area of paralleogram ABCD = AB × AE
AB = DCasoppositesideoftheparallelogramareequal
Therefore,
Area of paralleogram ABCD = 16 × 8 ……
1
Taking the base of Parallelogram ABCD as AD we get
Area of paralleogram ABCD = AD × CF
takingbaseasADandheightasCF
Area of paralleogram ABCD = AD × 10 ……
2
Since equation 1 and 2 both represent the Area of the same Parallelogram ABCD , both should be equal.
Hence fro equation
1 and
2,
This means that,
Hence we get the result as
Question:3
In Q.No. 1, if AD = 6 cm, CF = 10 cm, and AE = 8 cm, find AB.
Solution:
Given: Here in the question it is given that
1 ABCD is a parallelogram,
2 and
3 
4 AD = 6 cm
5 AE = 8cm
6 CF = 10cm
To Find : AB =?
Calculation: We know that formula for calculating the
Area of paralleogram = base × height
Therefore,
Area of paralleogram ABCD = DC × AE
TakingbaseasDCandHeightasAE
Area of paralleogram ABCD = AB × AE
AB = DCasoppositesideoftheparallelogramareequal
Therefore, Area of paralleogram ABCd = 16 × 8
Area of Parallelogram ABCD = AB× 8 ……
1
Taking the base of Parallelogram ABCD as AD we get
Area of paralleogram ABCD = AD × CF
takingbaseasADandheightasCF
Area of paralleogram ABCD = 6 × 10 ……
2
Since equation 1and 2 both represent the Area of the same Parallelogram ABCD , both should be equal.
Hence equation 1 is equal to equation 2
Which means that,
Hence we got the measure of AB equal to 
 
Question:4
Let ABCD be a parallelogram of area 124 cm
2
. If E and F are the mid-points of sides AB and CD respectively, then find
the area of parallelogram AEFD.
Solution:
Given: Here in the question it is given that
1 Area of paralleogram ABCD = 124 cm
2
2 E is the midpoint of AB, which means 
3 F is the midpoint of CD, which means 
To Find : Area of Parallelogram AEFD
Calculation: We know that formula for calculating the
Area of Parallelogram = base × height
Therefore,
Area of paralleogram ABCD = AB × AD
TakingbaseasABandHeightasAD ……
1
Therefore,
Area of paralleogram AEFD = AE × AD
TakingbaseasABandHeightasAD ……
2
( )
= Area of Parallelogram ABCD
fromequation1
Hence we got the result Area of Parallelogram AEFD 
 
Question:5
If ABCD is a parallelogram, then prove that
ar (?ABD) = ar (?BCD) = ar (?ABC) = ar (?ACD) = 
1
2
ar (||
gm
 ABCD)
Solution:
Given: Here in the question it is given that
1 ABCD is a Parallelogram
To Prove :
1 
2 
3 
4 
Construction: Draw 
Calculation: We know that formula for calculating the
Area of Parallelogram = base × height
Area of paralleogram ABCD = BC × AE
TakingbaseasBCandHeightasAE……(1
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Question:2
If the given figure, ABCD is a parallelogram, AE ? DC and CF ? AD. If AB = 16 cm. AE = 8 cm and CF = 10 cm, find AD.
 
Solution:
Given: Here in the question it is given
1 ABCD is a parallelogram,
2 and
3 , AB = 16 cm
4 AE = 8cm
5 CF = 10cm
To Find : AD =?
Calculation: We know that formula for calculating the
Therefore,
Area of paralleogram ABCD = DC × AE
TakingbaseasDCandHeightasAE
Area of paralleogram ABCD = AB × AE
AB = DCasoppositesideoftheparallelogramareequal
Therefore,
Area of paralleogram ABCD = 16 × 8 ……
1
Taking the base of Parallelogram ABCD as AD we get
Area of paralleogram ABCD = AD × CF
takingbaseasADandheightasCF
Area of paralleogram ABCD = AD × 10 ……
2
Since equation 1 and 2 both represent the Area of the same Parallelogram ABCD , both should be equal.
Hence fro equation
1 and
2,
This means that,
Hence we get the result as
Question:3
In Q.No. 1, if AD = 6 cm, CF = 10 cm, and AE = 8 cm, find AB.
Solution:
Given: Here in the question it is given that
1 ABCD is a parallelogram,
2 and
3 
4 AD = 6 cm
5 AE = 8cm
6 CF = 10cm
To Find : AB =?
Calculation: We know that formula for calculating the
Area of paralleogram = base × height
Therefore,
Area of paralleogram ABCD = DC × AE
TakingbaseasDCandHeightasAE
Area of paralleogram ABCD = AB × AE
AB = DCasoppositesideoftheparallelogramareequal
Therefore, Area of paralleogram ABCd = 16 × 8
Area of Parallelogram ABCD = AB× 8 ……
1
Taking the base of Parallelogram ABCD as AD we get
Area of paralleogram ABCD = AD × CF
takingbaseasADandheightasCF
Area of paralleogram ABCD = 6 × 10 ……
2
Since equation 1and 2 both represent the Area of the same Parallelogram ABCD , both should be equal.
Hence equation 1 is equal to equation 2
Which means that,
Hence we got the measure of AB equal to 
 
Question:4
Let ABCD be a parallelogram of area 124 cm
2
. If E and F are the mid-points of sides AB and CD respectively, then find
the area of parallelogram AEFD.
Solution:
Given: Here in the question it is given that
1 Area of paralleogram ABCD = 124 cm
2
2 E is the midpoint of AB, which means 
3 F is the midpoint of CD, which means 
To Find : Area of Parallelogram AEFD
Calculation: We know that formula for calculating the
Area of Parallelogram = base × height
Therefore,
Area of paralleogram ABCD = AB × AD
TakingbaseasABandHeightasAD ……
1
Therefore,
Area of paralleogram AEFD = AE × AD
TakingbaseasABandHeightasAD ……
2
( )
= Area of Parallelogram ABCD
fromequation1
Hence we got the result Area of Parallelogram AEFD 
 
Question:5
If ABCD is a parallelogram, then prove that
ar (?ABD) = ar (?BCD) = ar (?ABC) = ar (?ACD) = 
1
2
ar (||
gm
 ABCD)
Solution:
Given: Here in the question it is given that
1 ABCD is a Parallelogram
To Prove :
1 
2 
3 
4 
Construction: Draw 
Calculation: We know that formula for calculating the
Area of Parallelogram = base × height
Area of paralleogram ABCD = BC × AE
TakingbaseasBCandHeightasAE……(1
We know that formula for calculating the
Area of ?ADC = Base × Height
ADisthebaseof?ADCandAEistheheightof?ADC
= Area of Parallelogram ABCD
fromequation1
Hence we get the result 
Similarly we can show that 
2 
3 
4