8 MATHEMATICS. 
will the triangle acd be similar to abe. And if the second triangle, abu 
thus formed, is also to be similar to the original triangle, then must g= 0, 
also n-+-g or bac =m + 0, which is only possible when bac is a right angle. 
In this case the bisecting line ad is perpendicular to bc. If, therefore, in a 
right angled triangle, we let fall a perpendicular from the vertex of the 
right angle upon the hypothenuse, the perpendicular thus let fall will divide 
the triangle into two smaller ones, similar to each other and to the original 
triangle (jig. 39-41). From this may be easily deduced, 1, that the 
perpendicular let fall from the vertex of the right angle, is a mean propor- 
tional between two segments of the hypothenuse ; 2, that either side about 
the right angle is a mean proportional between the whole hypothenuse and 
adjacent segments. From the latter proposition follows another: that 
when the sides of a right angled triangle are expressed in numbers, the 
square of the hypothenuse will be equal to the sum of the squares of the 
other two sides. 
With respect to the similarity of such rectilineal figures as have more 
than three sides, we will confine ourselves here to the following proposition : 
two figures are similar, when they can be divided by similarly situated 
diagonals into triangles which are similar each to each (pl. 8, fig. 42). 
Similarity of figures may also be applied to the solution of numerous 
problems of construction, of which we will here mention only one ;—to find 
a fourth proportional to the three given lines, a, b,c (fig. 48). This is a 
problem of the same importance in Geometry as the Rule of Three is in 
Arithmetic. 
5. OF THE EQUIVALENCE OF AREAS IN FIGURES. 
Figures are said to be equivalent when they occupy equal areas. In 
equality we combine similarity with equivalence. We must here premise 
that in triangles and parallelograms, some one side is assumed as the ground 
line or basis upon which the figure is supposed to rest, and that then the 
height or altitude is the perpendicular distance from this basis to the oppo- 
site side or angle. 
Two parallelograms are equivalent, when their bases and altitudes are 
equal (pl. 3, fig. 45-47). Here we may always consider them as erected 
upon the same base, and the opposite sides will then be in one and the 
same parallel; in which case, apart from the condition of equality or com- 
plete coincidence, three conditions, as represented in figs. 45, 46, 47, are 
possible. A triangle is always the half of a parallelogram of the same base 
and altitude, therefore equal to a parallelogram of the same altitude and 
half the base, or to one of an equal base and half the altitude ; whence it 
follows that triangles of equal bases and altitudes are equivalent (jig. 48). 
If we assume in succession two different sides of the same triangle as bases, 
they will be inversely proportional to their corresponding altitudes, viz. 
ab: ac::be: cd (fig. 49). A trapezoid may be divided by a diagonal into two 
triangles, which will have the parallel sides of the trapezoid for their bases, 
and the perpendicular distance between these sides for their common alti- 
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