Similar polygons are polygons for which all corresponding angles are congruent and all corresponding sides are proportional.  Example: 

 

Many times you will be asked to find the measures of angles and sides of figures.  Similar polygons can help you out.

1. Problem: Find the value of x, y, and the measure of angle P.

Solution: To find the value of x and y, write proportions involving corresponding sides. Then use cross products to solve. 4 x 4 7 - = - - = - 6 9 6 y 6x = 36 4y = 42 x = 6 y = 10.5 To find angle P, note that angle P and angle S are corresponding angles. By definition of similar polygons, angle P = angle S = 86o.

Similar Triangles

Two triangles are similar if they have the same shape but different size.

Two triangles are similar if:

Their corresponding angles are equal.

Their corresponding sides are proportional, that is to say, they have the same ratio.

The perimeters of similar triangles have the same ratio.

The ratio of areas of similar triangles is equal to the square of the ratio.

Problems

1. Calculate the height of a building that casts a shadow of 6.5 meters if at the same

time and in the same place a pole of 4.5 m in height produces a shadow of 0.90 m.

2.The legs of a right triangle measure 24 m and 10 m. What is the length of the legs of

a similar triangle to this one whose hypotenuse is 52 m?

Similar Triangle Rules

1Two triangles are similar if two angles are equal.

 

2 Two triangles are similar if the sides are proportional.

 

3 Two triangles are similar if two sides are proportional and the angle between them is

equal.

Examples

Determine whether the following triangles are similar:

The triangles are similar because the sides are proportional.

180º − 100º − 60º = 20º

They are similar triangles because they have two equal angles.

They are similar because two sides are proportional and the angle between them is

equal.

Similar Right Triangles

1Two right triangles are similar if they have an equal acute angle.

2Two right triangles are similar if they have two legs that are proportional.

3Two right triangles are similar if the ratio of the hypotenuses and the legs are

proportional.

Definition:  Two triangles are similar if and only if the corresponding sides are in proportion and the corresponding angles are congruent.

There are three accepted methods of proving triangles similar:

AA To show two triangles are similar, it is sufficient to show that two angles of one triangle are congruent (equal) to two angles of the other triangle.

Theorem: If two angles of one triangle are congruent to two angles of another triangle, the triangles are similar. 

 

 

SSSfor similarity

BE CAREFUL!!  SSS for similar triangles is NOT the same theorem as we used for congruent triangles. To show triangles are similar, it is sufficient to show that the three sets of corresponding sides are in

proportion.Theorem: If the three sets of corresponding sides of two triangles are in proportion, the triangles are similar.

 

 

SASfor 

similarity

BE CAREFUL!!  SAS for similar triangles is NOT the same theorem as we used for congruent triangles.  To show triangles are similar, it is sufficient to show that two sets of corresponding sides are in

proportion and the angles they include are congruent.Theorem: If an angle of one triangle is congruent to the corresponding angle of another triangle and the lengths of the

sides including these angles are in proportion, the triangles are similar. 

 

Once the triangles are similar:

Theorem: The corresponding sides of similar triangles are in proportion. 

 

 

Dealing with overlapping triangles:

 

Many problems involving similar triangles have one triangle ON TOP OF  (overlapping) another triangle.  

Since   is marked to be parallel to  , we know that we have <BDE congruent to <DAC (by corresponding

angles).  <B is shared by both triangles, so the two triangles are similar by AA.

There is an additional theorem that can be used when working with overlapping triangles:AdditionalTheorem:

If a line is parallel to one side of a triangle and intersects the other two sides of the triangle, the line divides these two sides proportionally.

 

Similar Triangles

Two triangles are Similar if the only difference is size (and possibly the need to turn or flip one around).

These triangles are all similar:

(Equal angles have been marked with the same number of arcs)

Some of them have different sizes and some of them have been turned or flipped.

Similar triangles have:

all their angles equal

corresponding sides have the same ratio

Corresponding Sides

In similar triangles, the sides facing the equal angles are always in the same ratio.

For example:

Triangles R and S are similar. The equal angles are marked with the same numbers of arcs.

What are the corresponding lengths?

The lengths 7 and a are corresponding (they face the angle marked with one arc)

The lengths 8 and 6.4 are corresponding (they face the angle marked with two arcs)

The lengths 6 and b are corresponding (they face the angle marked with three arcs)

Calculating the Lengths of Corresponding Sides

It may be possible to calculate lengths we don't know yet. We need to:

Step 1: Find the ratio of corresponding sides in pairs of similar triangles.

Step 2: Use that ratio to find the unknown lengths.

Example: Find lengths a and b of Triangle S above.Step 1: Find the ratio

We know all the sides in Triangle R, and 

We know the side 6.4 in Triangle S

The 6.4 faces the angle marked with two arcs as does the side of length 8 in triangle R.

So we can match 6.4 with 8, and so the ratio of sides in triangle S to triangle R is:

6.4 to 8

Now we know that the lengths of sides in triangle S are all 6.4/8 times the lengths of sides in triangle R.Step 2: Use the ratio

a faces the angle with one arc as does the side of length 7 in triangle R.

a = (6.4/8) × 7 = 5.6

 

b faces the angle with three arcs as does the side of length 6 in triangle R.

b = (6.4/8) × 6 = 4.8

 Ratios and Proportions in Word Problems

Practice

1. If 5 pounds of grass seed will cover 1025 square feet, how many square feet can be covered by 15 pounds of

grass seed?

2. If a homeowner pays $8000 a year in taxes for a house valued at $250,000, how much would a homeowner pay

in yearly taxes on a house valued at $175,000?

3. If a person earns $2340 in 6 weeks, how much can the person earn in 13 weeks at the same rate of pay?

4. If 3 gallons of waterproofing solution can cover 360 square feet of decking, how much solution will be needed to

cover a deck that is 2520 square feet in size?

5. A recipe for 4 servings requires 6 tablespoons of shortening. If the chef wants to make enough for 9 servings,

how many tablespoons of shortening are needed?

Answers

1. 3075 square feet

2. $5600

3. $5070

4. 21 gallons

5. 13   or 13.5 tablespoons

Ratios and Proportions in Word Problems

In order to solve word problems using proportions:

Step 1 Read the problem .

Step 2 Identify the ratio statement .

Step 3 Set up the proportion .

Step 4 Solve for x .

In problems involving proportions, there will always be a ratio statement. It is important to find the ratio statement

and then set up the proportion using the ratio statement. Be sure to keep the identical units in the numerators and

denominators of the fractions in the proportion.

Examples

Example 1

An automobile travels 176 miles on 8 gallons of gasoline. How far can it go on a tankful of gasoline if the tank holds

14 gallons?

Solution 1

The ratio statement is:

so the proportion is:

Solve:

Hence, on 14 gallons, the automobile can travel a distance of 308 miles.

Math Note: Notice in the previous example that the numerators of the proportions have the same units, miles, and

the denominators have the same units, gallons.

Example 2

If it takes 16 yards of material to make 3 costumes of a certain size, how much material will be needed to make 8

costumes of that same size?

Solution 2

The ratio statement is:

The proportion is:

Hence,   yards or   yards should be purchased.

A ratio is a relationship between two values. For instance, a ratio of 1 pencil to 3 pens would imply that there are

three times as many pens as pencils. For each pencil there are 3 pens, and this is expressed in a couple ways, like

this: 1:3, or as a fraction like 1/3. There do not necessarily have to be those numbers of each, but a multiple of

them. We could just as easily have 2 pencils and 6 pens, 10 pencils and 30 pens, or even half a pencil and one-and-

a-half pens!

A proportion can be used to solve problems involving ratios. If we are told that the ratio of wheels to cars is 4:1, and

that we have 12 wheels, how can we find the number of cars we could have? A simple proportion will do perfectly.

We know that 4:1 is our given ratio and the new ratio with 12 wheels must be an equivalent fraction, so we can

setup the problem like this, where x is our missing number of cars:

To solve a proportion like this, we have to cross-multiply. This process involves multiplying the two extremes and

then comparing that product with the product of the means. An extreme is the first number (4), and the last

number (x), and a mean is the 1 or the 12.

To multiply the extremes we just do 4 * x = 4x. The product of the means is 1 * 12 = 12. The process is very simple

if you remember it as cross-multiplying, because you multiply diagonally across the equal sign.

You should then take the two products, 12 and 4x, and put them on opposite sides of an equation like this: 12 = 4x.

Solve for x by dividing each side by 4 and you discover that x = 3. Reading back over the problem we remember

that x stood for the number of cars possible with 12 tires, and that is our answer.

It is possible to have many variations of proportions, and one you might see is a double-variable proportion. It looks

something like this, but it easy to solve.

Using the same process as the first time, we cross multiply to get 16 * 1 = x * x. That can be simplified to 16 =

x^2, which means x equals the square root of 16, which is 4 (or -4). You've now completed this

 are built from ratios. A "ratio" is just a comparison between two different things. For instance, someone can look at a group of people, count noses, and refer to the "ratio of men to women" in the group. Suppose there are thirty-five people, fifteen of whom are men. Then the ratio of men to women is 15 to 20.

Notice that, in the expression "the ratio of men to women", "men" came first. This order is very important, and must be respected: whichever word came first, its number must come first. If the expression had been "the ratio of women to men", then the numbers would have been "20 to 15".

Expressing the ratio of men to women as "15 to 20" is expressing the ratio in words. There are two other notations for this "15 to 20" ratio:

odds notation:  15 : 20

fractional notation:  15/20

You should be able to recognize all three notations; you will probably be expected to know them for your test.

Given a pair of numbers, you should be able to write down the ratios. For example:

There are 16 ducks and 9 geese in a certain park. Express the ratio of ducks to geese in all three formats.

Consider the above park. Express the ratio of geese to ducks in all three formats.

The numbers were the same in each of the above exercises, but the order in which they were listed differed, varying according to the order in which the elements of the ratio were expressed. In ratios, order is very important.

Let's return to the 15 men and 20 women in our original group. I had expressed the ratio as a fraction, namely, 15/20. This fraction reduces to 3/4. This means that you can also express the ratio of men to women as 3/4, 3 : 4, or "3 to 4".

This points out something important about ratios: the numbers used in the ratio might not be theabsolute measured values. The ratio "15 to 20" refers to the absolute numbers of men and women, respectively, in the group of thirty-five people. The simplified or reduced ratio "3 to 4" tells you only that, for every three men, there are four women. The simplified ratio also tells you that, in any representative set of seven people (3 + 4 = 7) from this group, three will be men. In other words, the men comprise 3/7of the people in the group. These relationships and reasoning are what you use to solve many word problems:

In a certain class, the ratio of passing grades to failing grades is 7 to 5. How many of the36 students failed the course? Copyright © Elizabeth Stapel 2001-2011 All Rights Reserved

The ratio, "7 to 5" (or 7 : 5 or 7/5 ), tells me that, of every 7 + 5 = 12 students, five failed. That is, 5/12  of the class flunked. Then ( 5/12 )(36) = 15 students failed.

In the park mentioned above, the ratio of ducks to geese is 16 to 9. How many of the300 birds are geese?

The ratio tells me that, of every 16 + 9 = 25 birds, 9 are geese. That is, 9/25 of the birds are geese. Then there are ( 9/25 )(300) = 108 geese.

Generally, ratio problems will just be a matter of stating ratios or simplifying them. For instance:

Express the ratio in simplest form:  $10 to $45

This exercise wants me to write the ratio as a reduced fraction:

.10/45 = 2/9.

This reduced fraction is the ratio's expression in simplest fractional form. Note that the units (the "dollar" signs) "canceled" on the fraction, since the units, "$", were the same on both values. When both values in a ratio have the same unit, there should generally be no unit on the reduced form.

Express the ratio in simplest form: 240 miles to 8 gallons

 

 

When I simplify, I get (240 miles) / (8 gallons) = (30 miles) / (1 gallon), or, in more common language, 30 miles per gallon.