Unit 5 Review

Unit 5 Review

5.1 Inequalities and Triangles

5.2 Ratios, Proportions, Geometric Mean

5.3 Use Proportions in Geometry

5.4 Use Similar Polygons

5.5 Use Proportionality Theorems

5.6 Similarity Transformations

5.1 Inequalities and Triangles

• Not every set of 3 numbers can form a triangle

• Any two sides of a triangle MUST be LARGER than the third

• A very handy tool to visualize this can be found here: https://www.geogebra.org/m/pNm33AuP

Multiple Select

Which of the following groups of side lengths could NOT create a triangle?

5.2 Ratios

• A Ratio is a comparison of how much there is of one thing in relation to another; below is an example of different ratios within a single data group

• There are 10 students; 6 are left-handed while 4 are right-handed:

• The ratio of left-handed to right-handed students is 6 to 4, or 6:4

• The ratio of right-handed to left-handed students is 4 to 6, or 4:6

• The ratio of left-handed students to all students is 6 to 10, or 6:10

• The ratio of right-handed students to all students is 4 to 10, or 4:10

5.2 Simplifying Ratios

• Ratios can be simplified much like fractions

•  $\frac{2}{4}$  

• Is the above a proper answer to a math problem? NO!

•  $\frac{1}{2}$  

• That's better! Two-fourths simplifies to one-half

• A ratio can be simplified when both terms share a common factor - an integer that can be multiplied by another whole number to make the starting number

• If the terms share a common factor, we factor it out just as we do with a fraction. The ratio 2:4 simplifies to 1:2

Multiple Choice

What is the largest number that will fit into both 10 and 30? This is called the greatest common factor or (GCF)

• 5

• 3

• 10

• 4

Multiple Choice

What is the MOST simplified (absolutely smallest possible numbers) version of the ratio 10:30?

• 5:15

• 4:12

• 6:24

• 1:3

5.2 Proportionality

• If two ratios are proportional, they simplify to the same ratio

• An example of proportional ratios would be a scaled picture, model, or map

• The ratio of each component to all the others should be the same as in reality!

• These would then be proportional ratios, and proportional pictures, models, or map

Multiple Choice

This is Jamar. Hi Jamar!

Select the stick figure that is PROPORTIONAL to Jamar.

Multiple Choice

Which of the following ratios is proportional to the ratio 5:7?

• 10:21

• 3:7

• 7:5

• 15:21

5.2 Geometric Mean

• In Algebra 1 you learned about mean, median, and mode.

• The mean that you learned about, also known as the average, is called the arithmetic mean.

• The arithmetic mean is found by adding all of the values together and dividing by the total number of values.

• For example, the mean of 2, 4, 6 is found by the equation: $\frac{2\ +\ 4\ +\ 6}{3}=\frac{12}{3}=4$  

• The Geometric mean works similarly, but is different in a few specific, and important, ways.

Multiple Choice

Find the Algebraic Mean of 3, 5, and 8:

• 6

• 16

• 3

• 5

5.2 Geometric Mean

• If we have a set of n numbers (where n just represents how many numbers we have):

• The geometric mean is found by multiplying all of the values together, and then finding the nth root of the product.

• Remember that are not finding geometric means of more than two numbers! For these problems, always use the square root! √

• Find the geometric mean of 2 and 8:

•  $\sqrt{2\ \cdot\ 8}=\sqrt{16}=4$  

• Note how this is different from the arithmetic mean:

•  $\frac{2\ +\ 8}{2}=\frac{10}{2}=5$  

Multiple Choice

Find the geometric mean of 16 and 4:

• 8

• 10

• 12

• 20

Multiple Choice

Find the geometric mean of 4 and 36:

• 8

• 10

• 12

• 20

5.3 Use Proportions

• When we know that two things are proportional, we know that the ratio between those two simplify to the same values.

• We can create an equation that sets proportional ratios equal to each other, and solve for a missing value! Remember that we cross-multiple to solve.

• Let's return to our left-handed and right-handed students example. There are 10 students, 6 left-handed and 4 right-handed

• If I gain some students, and now have 12 left-handed students, but the ratio stays proportional, how right-handed students do I have? How many total?

•  $\frac{6}{4}=\frac{12}{x}\ \ \ \ =>\ \ \ 6x\ =\ 12\cdot4\ \ \ =>\ \ \ x\ =\ \frac{48}{6}\ \ \ =>\ \ x=8$ 

•  $\frac{6}{10}=\frac{12}{x}\ \ =>\ \ 6x\ =\ 12\ \cdot\ 10\ =>\ \ x\ =\ \frac{120}{6}\ \ =>\ \ x\ =\ 20$  

• 8 right-handed students and 20 total students!

5.3 Similarity and Corresponding Parts

• Two object are similar when they have the same shape but not necessarily the same side.

• Similar figures will have proportional ratios

• One of the most important skills in using proportions in geometry is the ability to find corresponding parts of figures

• Corresponding parts are those that are in the same relative place on different figures

• Thinking back to our stick figures - corresponding parts would be body parts! The heads correspond to each other - the right arms correspond to each other - etc.

Multiple Choice

Find the corresponding angle to angle a:

Multiple Choice

Find the side corresponding to side fg:

 $\overline{fg}$  

Use Proportions in Geometry

• With similar polygons, we can create ratios between sides, and compare those ratios with the corresponding ratios in similar figures.

• We can set up proportional relationships between corresponding parts of similar figures, just as we set up proportional relationships with the numbers of students.

• Then we can solve for missing components!

Multiple Choice

What is the ratio from side ab to side bc?

 $\overline{ab}$ to  $\overline{bc}$  
Remember that order is important!

• 6:4

• 4:5

• 5:6

• 6:5

Multiple Choice

What is the ratio from side ab to side bc?

 $\overline{de}$ to  $\overline{ef}$  
Remember that order is important!

• x:8

• 8:12

• 12:x

• x:12

Multiple Choice

What is the proportion of the following ratios:

 $\frac{\overline{ab}}{\overline{bc}}=\frac{\overline{de}}{\overline{ef}}$  

•  $\frac{4}{5}=\frac{8}{12}$  

•  $\frac{6}{5}=\frac{12}{x}$  

•  $\frac{6}{5}=\frac{x}{12}$  

•  $\frac{x}{8\ }=\frac{12}{x}$  

Fill in the Blanks

5.4 Similar Polygons

• For this section we focus on the idea of scale factor

• The scale factor is the number that one similar figure's values can be multiplied by to get the values of another figure

• Returning to the student example: if my classroom goes for 10 students to 20 students, what is the scale factor?

• To find the scale factor, divide the end value by the beginning value: in this case, divide 20 by 10. The scale factor is 2.

• What is my class goes from 20 students down to 4? Divide the end value (4) by the beginning value (20): 20 divided by 4 is 1/5. The scale factor is 1/5.

• When we know the scale factor between similar figures, we can find missing values.

Fill in the Blanks

Multiple Choice

If the scale factor between these similar figures is 5, what are the side lengths of figure b?

• 24

• 5

• 30

• 65

5.5 Proportionality Theorems (Triangles)

• We learned about two proportionality theorems: Side-splitter and Angle bisector

• The side-splitter theorem states that, in a triangle with an interior line parallel to one side of the triangle, the sides split by that line are proportional

• The angle-bisector theorem that, when a line in a triangle bisects one of the interior angles, the segments it creates are proportional to the adjacent sides.

Multiple Choice

Select the picture showing the side-splitter theorem:

Multiple Choice

Select the picture showing the angle bisector theorem:

5.6 Similarity Transformations

• In this section we learned about dilations, changing the size of a figure in relation to a specific point.

• To dilate a figure, we measure the distances of each point from the center of dilation and then multiply that distance by the dilation factor

• The resulting numbers are then added to the coordinates of the center of dilation to find the new location of the point

Multiple Choice

Which of the following is a dilation of point A from the origin (0,0) by a dilation factor of 2?

Multiple Choice

Which of the following is a dilation of point A from the origin (0,0) by a dilation factor of 1/2?

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