Triangle on same base and between same parallels is equal in area.
In the adjoining figure, ∆ABD and ∆DEF are having equal base ‘a cm’ and are between the same parallels BF and AD.
Therefore, area of ∆ABD = Area of ∆DEF
Prove that the triangles on same base and between same parallels are equal in area.
Let ∆ABC and ∆ABD be on the same base AB and between the same parallel AB and CD. It is require to prove that ∆ABC = ∆ABD.
Construction: A parallelogram ABPQ is constructed with AB as base and lying between the same parallels AB and CD.
Proof: Since ∆ABC and parallelogram ABPQ are on the same base AB and between the same parallels AB and Q,
Therefore, ∆ABC = ½(Parallelogram ABPQ)
Similarly, ∆ABD = ½(Parallelogram ABPQ)
Therefore, ∆ABC = ∆ABD.
Note: Since the relationship between the areas of a triangle and a parallelogram on the same base and between the same parallels in known to us, so that parallelogram ABPQ is constructed]
Solved examples for the triangle on same base and between same parallels:
1. Shaw that the medians of the triangle divide it into triangles of equal area.
Solution:
AD is the median of the ∆ABC and AE is the altitude of ∆ABC and also ∆ADC.
(AE ┴ BC)
AD is the median of ABC
Therefore, BD = DC
Multiply both sides by AE,
Then BD × AE = DC × AE
1/2 BD × AE = 1/2 DC × AE
Area of ∆ABD = Area of ∆ADC
2. AD is the median of ∆ABC and ∆ADC. E is any point on AD. Show that area of ∆ABE = area of ∆ACE.
Solution:
Since, AD is the median of ∆ABC, therefore BD = DC
Since, ∆ABD and ∆ADC have equal bases BD = DC and are between the same parallels BC and l,
Therefore Area of ∆ABD = Area of ∆ADC
Since, E lies on AD,
Therefore, ED is the median of the BEC
Now, BED and CED have equal bases BD = DC and between the same parallels BC and m.
Therefore, area of ∆BED = Area of ∆CED
On subtracting (1) and (2), we get
Area of ∆ABD  Area of ∆BED = Area of ∆ACD  Area of ∆CED
Area of ∆ABE = Area of ∆ACE
Figure on Same Base and between Same Parallels
Parallelograms on Same Base and between Same Parallels
Parallelograms and Rectangles on Same Base and between Same Parallels
Triangle and Parallelogram on Same Base and between Same Parallels
Triangle on Same Base and between Same Parallels
8th Grade Math Practice
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