Triangles on the same Base and between the same Parallels
Let us look at Fig. 9.18. In it, you have two triangles ABC and PBC on the same base BC and between the same parallels BC and AP.
What can you say about the areas of such triangles?
To answer this question, you may perform the activity of drawing several pairs of triangles on the same base and between the same parallels on the graph sheet and find their areas by the method of counting the squares.
Each time, you will find that the areas of the two triangles are (approximately) equal.
This activity can be performed using a geoboard also. You will again find that the two areas are (approximately) equal.
To obtain a logical answer to the above question, you may proceed as follows:
In Fig. 9.18, draw CD || BA and CR || BP such that D and R lie on line AP(see Fig.9.19).
From this, you obtain two parallelograms PBCR and ABCD on the same base BC and between the same parallels BC and AR.
`color{green}("Therefore,") "ar (ABCD) = ar (PBCR)"`
`color{blue}("Now") DeltaABC ≅ DeltaCDA and DeltaPBC ≅ DeltaCRP)`
`color{orange}(So,) ar (ABC) 1/2 ar and ar (PBC) = 1/2 ar (PBCR))`
`color{orange}("Therefore,") ar (ABC) = ar (PBC)`
In this way, you have arrived at the following theorem:
`color{red}("Theorem 9.2 : ")`
`bb"Two triangles on the same base (or equal bases) and between the same parallels are equal in area."`
Now, suppose ABCD is a parallelogram whose one of the diagonals is AC (see Fig. 9.20). Let AN⊥ DC. Note that
`DeltaADC ≅ Delta CBA)`
`color{red}(So,) ar (ADC) = ar (CBA))`
`color{navy}(Therefore,) ar (ADC) = 1/2 ar (ABCD)`
`- 1/2 (DC xx AN))`
So, area of `DeltaADC = 1/2 xx base DCxx "corresponding altitude" AN`
In other words, area of a triangle is half the product of its base (or any side) and the corresponding altitude (or height). Do you remember that you have learnt this formula for area of a triangle ?
From this formula, you can see that two triangles with same base (or equal bases) and equal areas will have equal corresponding altitudes For having equal corresponding altitudes, the triangles must lie between the same parallels.
From this, you arrive at the following converse of Theorem 9.2 .
`color{navy}("Theorem 9.3 :")`
`bb"Two triangles having the same base (or equal bases) and equal areas lie between the same parallels."`
Let us now take some examples to illustrate the use of the above results.
What can you say about the areas of such triangles?
To answer this question, you may perform the activity of drawing several pairs of triangles on the same base and between the same parallels on the graph sheet and find their areas by the method of counting the squares.
Each time, you will find that the areas of the two triangles are (approximately) equal.
This activity can be performed using a geoboard also. You will again find that the two areas are (approximately) equal.
To obtain a logical answer to the above question, you may proceed as follows:
In Fig. 9.18, draw CD || BA and CR || BP such that D and R lie on line AP(see Fig.9.19).
From this, you obtain two parallelograms PBCR and ABCD on the same base BC and between the same parallels BC and AR.
`color{green}("Therefore,") "ar (ABCD) = ar (PBCR)"`
`color{blue}("Now") DeltaABC ≅ DeltaCDA and DeltaPBC ≅ DeltaCRP)`
`color{orange}(So,) ar (ABC) 1/2 ar and ar (PBC) = 1/2 ar (PBCR))`
`color{orange}("Therefore,") ar (ABC) = ar (PBC)`
In this way, you have arrived at the following theorem:
`color{red}("Theorem 9.2 : ")`
`bb"Two triangles on the same base (or equal bases) and between the same parallels are equal in area."`
Now, suppose ABCD is a parallelogram whose one of the diagonals is AC (see Fig. 9.20). Let AN⊥ DC. Note that
`DeltaADC ≅ Delta CBA)`
`color{red}(So,) ar (ADC) = ar (CBA))`
`color{navy}(Therefore,) ar (ADC) = 1/2 ar (ABCD)`
`- 1/2 (DC xx AN))`
So, area of `DeltaADC = 1/2 xx base DCxx "corresponding altitude" AN`
In other words, area of a triangle is half the product of its base (or any side) and the corresponding altitude (or height). Do you remember that you have learnt this formula for area of a triangle ?
From this formula, you can see that two triangles with same base (or equal bases) and equal areas will have equal corresponding altitudes For having equal corresponding altitudes, the triangles must lie between the same parallels.
From this, you arrive at the following converse of Theorem 9.2 .
`color{navy}("Theorem 9.3 :")`
`bb"Two triangles having the same base (or equal bases) and equal areas lie between the same parallels."`
Let us now take some examples to illustrate the use of the above results.
