module similarities

8/19/2019 Module Similarities

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Module

in

MATH 8B

(ANALYTIC GEOMETRY)

SIMILARITY


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1.

Ratio and Proportion

1.1 Definition: If a and b are any two quantities of the same kind and the same units, the ratio of

a to b, written as a/b or a:b, is the result of comparing them by division.

Example: Find the ratio of a to b if a = 6 cm and b = 8 cm.

1.2 Definition: In the equal ratios

, two sequences of numbers and ,

where none of the numbers in the second sequence is 0, are said to be proportional.

Example: The sequence 2, 4, 6, … and 3, 6, 9, … are proportional since

1.3 Definition: A proportion is a statement of the equality of two ratios; in other words, the

statement a/b = c/d is a proportion.

Example: If a = 5, b = 12, c = 15 and d = 36 then a, b, c, and d form a proportion since

Exercises 1:1.1 Two segments AB and CD are in the ratio 3:5. If CD is longer than AB by 28 cm, how

long is each segment?1.2 The sides of a quadrilateral are in the ratio 2:3:3:4. If the perimeter is 72 cm, find the

length of each side.

1.3 The ratio of the areas of two triangles is 4:7. If the areas differ by 36cm2, find the area of

each triangle.

1.4 The supplement of an angle is 10° less than 3 times its complement. Find the reatio of the

complement to the supplement.

2.

Some Properties of Proportions

2.1 Property 1: In a proportion, the product of the means equals the product of the extremes.

(Product of means equals product of the extremes)

Example: Given

, since , the two ratios are equal and so

is a true proportion.2.2 Definition: If a, b, and c are three numbers such that a:b = b:c, then b is the geometric mean

or mean proportional between a and c.

Example: In the proportion 4:6 = 6:9, 6 is the geometric mean or the mean proportional

between 4 and 9.

2.3 Property 2: The mean proportional or geometric mean of two numbers is the square root of

their product. (Geometric mean of two numbers.)

Examples: If a = 4 and c = 8, what is the geometric mean of a and c?

2.4 Property 3: If the ratios of a proportion are inverted, the result is still a proportion, that is if

a:b = c:d, then b:a = d:c is also a proportion. (Inversion property of proportions)

Example: 3:4 = 9:12 is a proportion, so is 4:3 = 12:9

2.5 Property 4: Interchanging the means or extremes of a proportion results in other proportion,

that is, if

are both proportions. (Exchange of means and

extremes property)

Example: 3:4 = 9:12 is a proportion, so is 12:4 = 9:3.


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2.6 Property 5: Adding 1 to or subtracting 1 from both sides of a proportion results in another

proportion, that is if

, then

.

Example: Since 7:5 = 21:15, then

are both

proportions.

2.7

Property 6: If

then

.

Example: Since

, then

.

Exercises 2:

2.1 In

i. If ii. If iii. If iv. If

2.2 i. In

Find the lengths of AB and AC

if the measurements are in centimeters.

ii. If

and , for the same value of x in (i),

find the lengths of XY and BC.

2.3 If , find two possible values of the ratio a:b

2.4 Generalized Property 6, that is, if

., prove that

3

Areas of Triangles

3.1

Theorem 7.1: (Ratio of the Areas of Two ). The ratio of the areas of two triangles is

equal to the ratio of products of their altitude and bases. Example: Suppose that in

What is What is h:a?

Sol.

3.2 Theorem 7.2: (Ratio of with equal bases): If two triangles have equal bases, the ratioof their areas equals the ratio of their altitudes.

Example: In the example above, you can barely conclude that

3.3

Theorem 7.3: (Ratio of Triangles with Equal Altitudes): If two triangles have equalaltitudes, then the ratio of their areas equals the ratio of their bases.

Example: Suppose that in

What is What is : ?

Sol.


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3.4 Theorem 7.4: (Areas of between parallel lines): Triangles on the same or equal basesand between the same parallels have equal areas.

3.5 Theorem 7.5: ( with Equal Areas and Bases): If two triangles have equal areas andthey have the same or equal bases, then their altitudes are equal.

3.6

Corollary 7.6: If two triangles have equal areas and the same or equal bases lying on the

same line, then the vertices opposite the bases lie on a line parallel

Exercises 3:

3.3

3.4.

4

The Basic of Proportionality

4.1 Definition: Two line segments are divided proportionally when the ratio of the lengths

of the segments into which one is divided is equal to the ratio of the lengths of the

corresponding segments into which the other is divided.


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4.2 Theorem 7.7: (Basic Proportionality Theorem): If a line parallel to one side of a

triangle intersects the other two sides, then it divides them proportionally.

4.3 Theorem 7.8 (Converse of the Basic Proportionality Theorem): If a line divides two

sides of a triangle proportionally, the line is parallel to the third side.

4.4 Theorem 7.9 (Proportional Segments Theorem): Parallel lines cut proportional

segments on all transversals.

4.5 Theorem 7.10 (Bisector of an Angle of a Triangle Theorem): The bisector of an angle

of a triangle divides the opposite side into segments proportional to the other two sides

to which they are adjacent.


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Exercises 4:

5.


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5 Similar Polygons

5.1 Definition: Two polygons are similar if: a) corresponding angles are equal and b)the

ratios of all pairs of corresponding sides are equal

5.2 Definition: If two polygons are similar, the ratio of the lengths of any pair of

corresponding sides is called the ratio of similitude.

Exercises 5:

1.


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2.

3.

6 The AAA Similarity Theorem

6.1 Theorem 7.11 (AAA Similarity Theorem): Two triangles with corresponding angles

equal are similar.

6.2 Corollary 7.12 (AA Corollary): Two Triangles are similar if two angles of one are equal

to the corresponding angles of the other.6.3 Corollary 7.13 (Acute Angle Similarity Corollary): Two right triangles are similar if an

acute angle of one is equal to an acute angle of another.

Exercises 6: 1. .

2.


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.

3.

.

4. .

7 The SAS and SSS Similarity Theorems

7.1 Theorem 7.14 (SAS Similarity Theorem): If t wo sides of one triangle are proportional

to two sides of another and the included angles are equal, then the triangles are similar.

7.2 Theorem 7.15 (SSS Similarity Theorem): If three sides of one triangle are proportional

to the corresponding sides of another, then the two triangles are similar.

Exercises 7:

1. In each figure, find the unknown lengths. Arrows indicate parallelism.

2. In the figure, DC||FE||AB

3. .


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8 Special Right Triangles

8.1 Theorem 7.16 (Right Triangle Similarity Theorem): In a right triangle, the altitude to

the hypotenuse divides the triangle into two triangles each of which is similar to the

original triangle.

8.2 Corollary 7.17 (Geometric Mean Theorem): In a right triangle, (a) the altitude to the

hypotenuse is the geometric mean to the segments into which it divides the hypotenuse;(b) each leg is geometric mean of the hypotenuse and the segment of the hypotenuse

adjacent to the leg.

8.3 Theorem 7.18 (Pythagorean Theorem): In a right triangle, the square of the

hypotenuse is equal to the sum of the squares of the two legs.


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8.4 Theorem 7.19 (Converse of the Pythagorean Theorem): In any triangle, if the sum of

the squares of two sides is equal to the squares of the third side, then the triangle is a

right triangle.

8.5 Theorem 7.20: In a 30°-60°-90° Triangle, the shorter leg is half the length of the

hypotenuse and the longer leg is

of the length of the hypotenuse.

8.6 Theorem 7.21: The length of the hypotenuse of an isosceles right triangle is times thelength of a side

Exercises 8:


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B. 1

5..

5.

6. .

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