can be applied in several forms and has applications in several places. We would discuss some forms of applications and then application of proportions
in similar triangles
Direct Proportion/ Direct Variation
If two quantities changes or vary in such a manner that their ratio
always remains constant, the quantities are said to be in proportion
. In other words, the two quantities are related in such a manner that the positive change in one quantity leads to proportionately same positive change in the other quantity. We represent this proportionality using 'α' (
). Thus x α y is termed as 'x is directly proportional to y'.
If y α x then y = kx , where k is a non-zero constant called constant of proportionality
. The equation is called the equation of direct proportionality
Inverse proportion/ Inverse Variation
If two quantities changes or vary in such a manner that an increase or decrease in one quantity leads to the proportional decrease or increase respectively in the other quantity, they are said to be in inverse proportion
If y α (1/x), then y = k/x where k is again a constant of proportionality
and a non zero constant.
In this we have both types of variations, direct and indirect.
If x α (1/y) and x α z, then combining them we can say
x α (z/y) then x =((k*z)/y).
Some more results on proportions:
If a α b and b α c then a α c
If a α b and c α d then ac α bd
If a α b then ap α bp, where b is the constant of proportionality.
If a α b then a^n α b^ n.
If a α b and c α b then (a (+, -) c ) α b .
Application of proportion in similar triangles
have a property that the corresponding side are in proportion
. If ΔABC is similar to ΔXYZ, then corresponding sides are
AB -> XY
BC -> YZ
CA -> ZX
These sides are in proportion
If we know 4 of these sides, we can calculate the remaining two sides using the two equations which can be formed.
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