Lesson Applications of Proportions

Source code of 'Applications of Proportions'

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<a href="proportions_intro.lesson">Proportions</a> can be applied in several forms and has applications in several places. We would discuss some forms of applications and then application of <a href="proportions_intro.lesson">proportions</a> in <a href="Similarity_(geometry).wikipedia">similar triangles</a>.

Direct Proportion/ Direct Variation
If two quantities changes or vary in such a manner that their <a href="ratios.lesson">ratio</a> always remains constant, the quantities are said to be in <a href="proportions_intro.lesson">proportion</a>. 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 '&#945;' ({{{alpha}}}). Thus x &#945; y is termed as 'x is directly proportional to y'.

If y &#945; 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 <a href="proportions_intro.lesson">proportion</a>.

If y &#945; (1/x), then y = k/x where k is again a constant of proportionality and a non zero constant.

Mixed Variation
In this we have both types of variations, direct and indirect. 
If x &#945; (1/y) and x &#945; z, then combining them we can say
x &#945; (z/y) then x =((k*z)/y).

Some more results on <a href="proportions_intro.lesson">proportions</a>:
If a &#945; b and b &#945; c then a &#945; c 
If a &#945; b and c &#945; d then ac &#945; bd
If a &#945; b then ap &#945; bp, where b is the constant of proportionality.
If a &#945; b then a^n &#945; b^ n.
If a &#945; b and c &#945; b then (a (+, -) c ) &#945; b .

Application of <a href="proportions_intro.lesson">proportion</a> in <a href="Similarity_(geometry).wikipedia">similar triangles</a>
<a href="Similarity_(geometry).wikipedia">Similar triangles</a> have a property that the corresponding side are in <a href="proportions_intro.lesson">proportion</a>. If &#916;ABC is similar to &#916;XYZ, then corresponding sides are
AB -> XY
BC -> YZ
CA -> ZX
These sides are in <a href="proportions_intro.lesson">proportion</a> would mean
{{{AB/XY = BC/YZ = CA/ZX}}}
If we know 4 of these sides, we can calculate the remaining two sides using the two equations which can be formed.