Understand how to find the area of a Parallelogram and area of a Triangle. click on the link to Watch the VIDEO explanation: Watch Video
Numerical
Let us solve the numerical on area theorems
Problem: ABCD is a parallelogram. Given that AM is equal to BM and area of triangle AMN is equal to 27 cm square. Find area of triangle ADM and parallelogram ABCD
Let us now identify the data given to us
ABCD is a parallelogram
Am is equal to BM
Area of triangle AMN is equal to 27 cm square
We need to find the area of triangle ADM and area of parallelogram ABCD
Calculation:
We need to do a construction before we proceed with the problem.
What do you think is the required construction?
Can you pick from the choices below
That's right. Join DB
Triangle AMN is equal to BMN Since They have equal bases AM and BM and same vertex N.
Therefore, Area of triangle BMN is equal to 27 cm square since area of triangle AMN is equal to 27 cm square
Area of triangle ANB is equal to Area of triangle AMN plus Area of triangle BMN
Calculate and key in the value in the text box given. Click on check to verify your answer
That's right. Therefore, Area of triangle ANB is equal to 54 cm square
In triangles ADM and BNM It is given that AM is equal to BM
Angle AMD is equal to angle BMN since they are the vertically opposite angles
Angle DAM is equal to angle NBM since they are the alternative angles and DA is parallel to BN
Therefore, by ASA congruency triangle ADM is equal to triangle BNM
Therefore, AD is equal to BN corresponding part of corresponding triangle
Therefore ANBD is a parallelogram. Since AD is parallel to BN and AD is equal to BN and AB is a diagonal of this parallelogram ANBD.
Therefore, Area of triangle ADB is equal to Area of triangle ANB is equal to 54 cm square.
DM is the median of triangle ADB, so divides triangle ADB into two triangles ADM and BDM of equal area
Calculate and key in value of area of triangle ADM . Click CHECK to verify your answer.
That's right Area of triangle ADM is equal to 27 cm square
Also, area of parallelogram ANBD is equal to 2 into Area of triangle ANB
2 into 54 which is equal to 108 cm square
Since parallelogram ABCD and parallelogram ANBD lie on the same base AD and between same parallels DA and CN, we have Area of parallelogram ABCD is equal to Area of parallelogram ANBD
Key in the value in the text box then find the area of triangle ABCD.
Yes you got it right.
Therefore area of parallelogram ABCD is equal to 108 cm square.
Numerical
Let us solve the numerical on area theorems
Problem: ABCD is a parallelogram. Given that AM is equal to BM and area of triangle AMN is equal to 27 cm square. Find area of triangle ADM and parallelogram ABCD
Let us now identify the data given to us
ABCD is a parallelogram
Am is equal to BM
Area of triangle AMN is equal to 27 cm square
We need to find the area of triangle ADM and area of parallelogram ABCD
Calculation:
We need to do a construction before we proceed with the problem.
What do you think is the required construction?
Can you pick from the choices below
That's right. Join DB
Triangle AMN is equal to BMN Since They have equal bases AM and BM and same vertex N.
Therefore, Area of triangle BMN is equal to 27 cm square since area of triangle AMN is equal to 27 cm square
Area of triangle ANB is equal to Area of triangle AMN plus Area of triangle BMN
Calculate and key in the value in the text box given. Click on check to verify your answer
That's right. Therefore, Area of triangle ANB is equal to 54 cm square
In triangles ADM and BNM It is given that AM is equal to BM
Angle AMD is equal to angle BMN since they are the vertically opposite angles
Angle DAM is equal to angle NBM since they are the alternative angles and DA is parallel to BN
Therefore, by ASA congruency triangle ADM is equal to triangle BNM
Therefore, AD is equal to BN corresponding part of corresponding triangle
Therefore ANBD is a parallelogram. Since AD is parallel to BN and AD is equal to BN and AB is a diagonal of this parallelogram ANBD.
Therefore, Area of triangle ADB is equal to Area of triangle ANB is equal to 54 cm square.
DM is the median of triangle ADB, so divides triangle ADB into two triangles ADM and BDM of equal area
Calculate and key in value of area of triangle ADM . Click CHECK to verify your answer.
That's right Area of triangle ADM is equal to 27 cm square
Also, area of parallelogram ANBD is equal to 2 into Area of triangle ANB
2 into 54 which is equal to 108 cm square
Since parallelogram ABCD and parallelogram ANBD lie on the same base AD and between same parallels DA and CN, we have Area of parallelogram ABCD is equal to Area of parallelogram ANBD
Key in the value in the text box then find the area of triangle ABCD.
Yes you got it right.
Therefore area of parallelogram ABCD is equal to 108 cm square.
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