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PROPORTIONS OF THE TRIANGLE

lule just pointed out. It is the rule of three; so namedbecause with the three terms of a proportional given, wemay find the fourth. The rule of three is continually usedin calculations of finance, of trade, and of business; andgeometry also has its rule of three.

If we have three lines (A) (B) (C), fig. 6, pi. 5, it is easy to finda fourth, D, so that we should have as follows.

(A) : (B) :: (C) : (D.)

Begin by making (C) = l’R at the end of A = OP. From theextremity of O draw the right line O M in any direction; fromthe point O, take the length O Q = (B), draw PQ, and draw RSparallel to PQ. Then we have

OP : OQ :: PR : QSOr(A) :(B):: (C): D.

When the two mean terms are equal to one another, thelength or the number which represents them is what iscalled the mean proportional between the extremes. Thus,in the proportion

2 : 4 : : 4 : 8

4 is the mean proportional between the two extremes 2and 8.

In geometry, two lines being given in length, their meanproportional is easily found. This will be hereafter ex-plained.

Similar triangles. —If two triangles ABC, abc , fig. 7,pi. 5, have their corresponding sides parallel, these sidesare proportional, and the triangles are similar. ThusAB : ab :: BC : be : : AC : ac.

To demonstrate this, let us move abc without changing the direc-tion of its sides, so that the point b falls on A; produce ac and BCtill they meet in a point m, and we shall have AC = cm, Cm = be,because they are parallels, comprised between parallels.

But AC and cm, Cm, and be, being parallel, we haveAB : ab : : cm = AC : acAB : ab :: BC : Cm = be

Whence AB : ab :: AC : ac :: BC : be.

If two triangles ABC, abc,, fig. 8, are so placed andformed, that AB shall be perpendicular to ab, BC to be,AC to ac, these two triangles are similar.