V. SIMILAR TRIANGLES

AB/DE = BC/EF = AC/DF = perimeter of ABC/ perimeter of DEF

Two triangles are similar if:

3 angles of 1 triangle are the same as 3 angles of the other
3 pairs of corresponding sides are in the same ratio
An angle of 1 triangle is the same as the angle of the other triangle and the sides containing these angles are in the same ratio.

RATIO

A ratio is a comparison of two numbers. We generally separate the two numbers in the ratio with a colon (:). Suppose we want to write the ratio of 8 and 12.
We can write this as 8:12 or as a fraction 8/12, and we say the ratio is eight to twelve.

Examples:

Jeannine has a bag with 3 videocassettes, 4 marbles, 7 books, and 1 orange.

1) What is the ratio of books to marbles? Expressed as a fraction, with the numerator equal to the first quantity and the denominator equal to the second, the answer would be 7/4. Two other ways of writing the ratio are 7 to 4, and 7:4.

2) What is the ratio of videocassettes to the total number of items in the bag? There are 3 videocassettes, and 3 + 4 + 7 + 1 = 15 items total. The answer can be expressed as 3/15, 3 to 15, or 3:15.

Comparing Ratios

To compare ratios, write them as fractions. The ratios are equal if they are equal when written as fractions.

Example:

Are the ratios 3 to 4 and 6:8 equal? The ratios are equal if 3/4 = 6/8. These are equal if their cross products are equal; that is, if 3 × 8 = 4 × 6. Since both of these products equal 24, the answer is yes, the ratios are equal.

Remember to be careful! Order matters! A ratio of 1:7 is not the same as a ratio of 7:1.

Examples:

Are the ratios 7:1 and 4:81 equal? No!
7/1 > 1, but 4/81 < 1, so the ratios can't be equal. Are 7:14 and 36:72 equal? Notice that 7/14 and 36/72 are both equal to 1/2, so the two ratios are equal.

PROPORTION

A proportion is an equation with a ratio on each side. It is a statement that two ratios are equal.
3/4 = 6/8 is an example of a proportion.

When one of the four numbers in a proportion is unknown, cross products may be used to find the unknown number. This is called solving the proportion. Question marks or letters are frequently used in place of the unknown number.

Example:

Solve for n: 1/2 = n/4. Using cross products we see that 2 × n = 1 × 4 =4, so 2 × n = 4. Dividing both sides by 2, n = 4 ÷ 2 so that n = 2.

BASIC PROPORTIONALITY THEOREM

If a line is drawn parallel to one side of a triangle and it intersects the other two sides at two distinct points then it divides the two sides in the same ratio.

CONVERSE OF BASIC PROPORTIONALITY THEOREM

If a line divides any two sides of a triangle in the same ratio, the line must be parallel to the third side.

BASIC SIMILARITY THEOREM

If three sides of a triangle are proportional to the three sides of another triangle, then the triangles are similar (SSS Similarity Theorem).

If the angles (two implies three) of two triangles are equal, then the triangles are similar (AA Similarity Theorem).

Note: this applies not only to ASA, AAS=SAA, but also to AAA situations.

If two sides of a triangle are proportional to two sides of another triangle and theincluded angles are congruent, then the triangles are similar (SAS Similarity Theorem).

Thus remains the SSA (ASS) case, which remains ambiguous unless HL or SsA occurs.

Lines parallel to a side of a triangle intersect the other two sides at nonvertices,
if and only if the two sides are split into proportional segments.

GEOMETRIC MEAN

We introduced the geometric mean somewhat in the last chapter and somewhat in statistics. Please review what we have there. The geometric mean is typically first encountered in a proportion when the means are equal, as in 8/w=w/4. Here w²=32 and square rooting both sides gives an answer. However, in general, there may be n n^th geometric means. We thus cannot be sure of the sign of w above.

The geometric mean is developed here because of its application to right triangles and the way the altitude to the hypotenuse divides the triangle into similar triangles. Assume you have two of the three terms in a geometric sequence, such as 2, ?, 50. In other words, you want some number g, such that 2/g=g/50, or 2•50=100=g², so obviously, g=10 or perhaps g=-10. Often the positive geometric mean is required and will be so specified.

The use of lower case letters a, b, and c for the sides of a triangle is a common convention dating back to Euler which we will adhere to. a refers either to the set of points composing the side or the length of the side, depending on context. The angle opposite side a is A, the angle opposite side b is B, and the angle opposite side c is C. If it is a right triangle, C will be right so c will be the [length of the] hypotenuse. Given the right triangle ABC with height h (CD) to the hypotenuse, h= (xy), whereas a= (cx), and b= (cy). Here x+y=c (BD + AD = AB), and y is the leg of a similar triangle with hypotenuse b, and x is the leg of a similar triangle with hypotenuse a.

The altitude of a triangle is the geometric mean of the segments of the hypotenuse that it divides.

Each leg would also be a geometric mean of the hypotenuse and the adjacent segment of the hypotenuse.

SPECIAL TRIANGLES

An isosceles right triangle (45^°–45^°–90^°) is a very special triangle. Its side lengths form a very special ratio which must be memorized. Specifically, if the legs are both of length x, then the hypotenuse is of length x by the pythagorean theorem. This ratio of 1/ = /2 or about 0.707 must become familiar.

Similarly, the 30^°–60^°–90^° triangle must be memorized, somehow. One way is to start with an equilateral triangle, bisect one angle which also bisects the side opposite, and consider the resulting congruent triangles. Obviously, two congruent 30^°–60^°–90^° triangles are formed. Again, by the pythagorean theorem, the side length ratios can be found to be 1: :2. By the AA Similarity Theorem, any triangle with these angles has these exact side length ratios.