GEOMETR Y CHAPTER 8 Similarity L8 1 RATIO

GEOMETR Y CHAPTER #8 Similarity

L#8. 1 RATIO & PROPORTION A ratio is a comparison of two numbers with the same units. A ratio should be written in reduced form.

Rate A rate is a comparison of two measurements with different units.

RATE A unit rate is a comparison of two measurements in which one of the terms has a value of 1. measurements which into a unit rate by making the Any rate can be in converted one of terms has a value of 1. numerator or the denominator equal to 1, but of course, the other number would have to change as well. Examples of unit rates: Using unit rates, I can find out how much it costs per song on the i. Tunes album. Unit rate Using unit rates, I can find out how far the cyclist went each hour. Unit rate

USING UNIT RATE Application Example: Meijer has Sprite on sale this week. You can buy a 1. 25 liter bottle for $0. 99, or you can get two 2 liter bottles for $3. 00. Which is the better buy? Unit rate It is a better buy to buy the two 2 liters for $3. 00 (as long as you have that much money).

PROPORTION A proportion is an equation that states that two ratios are equivalent. If one of the numbers in the proportion is unknown, you can SOLVE the proportion to find the missing value. One way to do this is mental math. x 3 Sometimes it is not easy to solve proportions with mental math. There’s another method you can use.

PROPORTION proportion is an equation that states that two ratios are equivalent. A second method to solve proportions is to get the variable alone. (12) x = 24 8 x = 3

PROPORTION proportion is an equation that states that two ratios are equivalent. A third method to solve proportions is to cross multiply. 8 x = 12 (2) 8 x = 24 8 8 x = 3

Terms of a Proportion The extremes of a proportion are the first and last terms. x and 8 are the extremes. The means of a proportion are the second and third terms. 12 and 2 are the means.

PROPERTIES OF PROPORTIONS

L#8. 2 MORE PROPERTIES OF PROPORTIONS

Geometric Mean The geometric mean of two positive numbers a and b is the positive number x such that Example: What is the geometric mean of 8 and 18? x = 12

WHAT DOES SIMILAR MEAN IN GEOMETRY? Two figures are SIMILAR FIGURES if they have THE SAME SHAPE BUT NOT NECESSARILY THE SAME SIZE.

L#8. 3

The SCALE FACTOR of two polygons is the reduced ratio of their corresponding sides

Theorem 8. 1 If two polygons are similar, similar then the ratio of their perimeters is equal to the ratios of their corresponding side lengths

L#8. 4 SIMILAR TRIANGLES Write a statement of proportionality. Find the measure of all unlabeled angles. Find the measure of missing side x. 12 x = 3(x + 15) 12 x = 3 x + 45 9 x = 45 x=5

L#8. 4 SIMILAR TRIANGLES Write a statement of proportionality. Find the measure of missing side y. 20 y = 5(12) 20 y = 60 y=3

L#8. 5 PROVING TRIANGLES SIMILAR In previous lessons, we have said that we can prove 2 triangles are similar if we can show that… 1) All their corresponding angles are congruent AND 2) All their corresponding sides are in proportion In this lesson, we will learn 2 theorems that we can use to prove triangles are similar.

Theorem 8. 2 Side-Side (SSS) Similarity If. Theorem the lengths of the corresponding sides of 2 triangles are proportional, then the triangles are similar.

Theorem 8. 3 Side-Angle-Side (SAS) Similarity Theorem If an angle of one triangle is congruent to an angle of a second triangle and the lengths of the sides including these angles are proportional, then the triangles are similar.

L#8. 6 PROVING TRIANGLES SIMILAR In previous lessons, we have said that we can prove 2 triangles are similar if we can show that… 1) All their corresponding angles are congruent AND 2) All their corresponding sides are in proportion In the last lesson, we learned 2 theorems that we could use to prove triangles are similar. In this lesson, we will learn a postulate that we can use to prove triangles are similar.

Postulate 25 Angle-Angle (AA) Similarity Theorem If 2 angles of one triangle are congruent to 2 angles of another triangle, then the two triangles are similar.

Summary of Ways to Prove Triangles are Similar