Name______Date______Hour______
9.1 – Using Ratios and Proportions J
A ______is a comparison of two quantities.
The ratio of a to b can be expressed as or or
Examples:
Write each ratio in simplest form-
1. 2. 3.
4. 5. six days to two weeks 6. 24 inches : 3 feet
7. 45 centimeters to 7 meters 8. 17 yards to 15 feet
9. 280 seconds : 6 minutes 10. 75 meters to 5 kilometers
A ______is an equation that shows two equivalent fractions.
There are three methods to determine if a ratio forms a proportion.
Method 1 Method 2 Method 3
Simplify the fractions Determine the decimals Cross Multiply
So, the answer is “YES” since the fractions, the decimals, and the cross product are equal.
Examples:
Determine whether the following are a proportion:
11. 12.
In the proportion below there are two cross-products.
11 and x ______
16 and 44 ______
You can use cross-multiplication to solve equations in proportion form…
Examples:
Solve each proportion by using cross-products.
1. 2. 3.
4. 5.
Geometry G Name ______
Ratios Worksheet 1
Express each ratio in lowest terms.
1. 8 to 16 ______2. 12 to 4 ______3. 15 : 75 ______
4. ______5. 150 to 15 ______6. ______
Write each ratio in lowest terms.
7. 15 milliliters to 24 liters ______8. 6 feet to 15 inches ______
9. 75 cm to 4 m ______10. 3 days to 9 hours ______
11. A soccer team played 25 games and won 17.
a. What is the ratio of the number of wins to the number of loses?
b. What is the ratio of the number of games played to the number of games won?
12. In a senior class, there are b boys and g girls. Express the ratio of the number of boys to the
total number in the class.
13. Two numbers are in a ratio of 5 : 3. Their sum is 80. Find the largest number.
14. Mr. Smith and Mr. Kelly are business partners. They agreed to divide the profits in the ratio of 3 : 2. The profit amounted to $24,000. How much did each person receive?
Geometry G Name ______
Ratios Worksheet 2 Period ______Date ______
Express each ratio in lowest terms.
1. ______2. 96 : 100 ______3. 625 to 125 ______
4. 72 to 60 ______5. ______6. 49 : 35 ______
7 15 kg to 90 kg ______8 18 feet to 4 yards ______
9. 45 meters to 80 meters ______10. 10 seconds to 2 minutes ______
11. The Yankees won 125 games, the Red Sox won 97 games, and the Mets won 86 games. What is the ratio of wins of the Yankees to the Red Sox to the Mets?
12. The measure of the angles of a triangle are in a ratio of 2 : 3 : 4. Find the number of degrees in the smallest angle of the triangle.
Do the following pairs form a proportion?
13. and 14. and 15. and
Geometry G Name ______
Ratios Worksheet 3 Period ______Date ______
Solve each proportion. Circle your final answer.
1. 2. 3.
4. 5. 6.
7. 8. 9.
Geometry G Name ______
Ratios Worksheet 4 Period ______Date ______
Applications of Proportions
1. A recipe for 3 dozen cookies calls for 4 cups of flour. How much flour is needed to make 5 dozen cookies? / 2. A certain medication calls for 250 mg for every 75 lbs of body weight. How many milligrams of medication should a 220-lb person take?3. A 2-inch wound requires 9 inches of suture thread. How long of a thread should a nurse have ready to close a 5-inch wound? / 4. An apartment building has 24 identical apartments. It took 42.7 gallons of paint to paint 3 apartments. How many gallons of paint are needed to paint 21 apartments?
Do the following ratios form a proportion? Meaning, are they equal?
1. 2. 3. 4.
Sect 9.2 - Changing the Size of Figures
These figures are similar These are not similar
Similar Figures ~ Two polygons are similar if and only if the______angles are ______and the measures of the ______sides are ______.
The symbol ______means similar.
DABC ~ DDEF (“triangle ABC is similar to triangle DEF”)
Corresponding Angles Corresponding Sides
are ______have ______
Ð ______ó Ð ______ó ______
Ð ______ó Ð ______ó ______
Ð ______ó Ð ______ó ______
Scale Factor -
If the scale factor > 1,
If the scale factor < 1,
Example: Find the dimensions of the figure ...
a) using a scale factor of 2. b) using a scale factor of .
Similar figures are enlargements or reductions of each other. The amount of enlargement or reduction needed to change one figure to the other is called the ______. The ratio of the lengths of the corresponding sides of similar figures is the ______.
Determine if the polygons are similar. Show work to justify your answer.
1) 2) 3)
Find the values of x and y if DJHI~DMLN.
a) Write proportions for the corresponding sides.
b) Write the proportion c) Write the proportion
to solve for x. to solve for y.
Example: ABCD is similar to WXYZ
The similarity ratio of ABCD to WXYZ is ______.
The scale factor of ABCD to WXYZ is ______.
Label the lengths of the missing sides.
ABCDE is similar to QRSTU
The similarity ratio of ABCDE to QRSTU is ______.
The scale factor of ABCDE to QRSTU is ______.
Find the length of each side.
QU ______
QR ______
RS ______
ST ______
Perimeter of ABCDE______
Perimeter of QRSTU______
ratio of perimeter of ABCDE to perimeter of QRSTU ______
Geometry Name
Chapter 11.1 Scale Factor Worksheet 1
Scale factor of 3
Scale factor of 2/3
Scale factor of 3/4
Geometry Name
Chapter 11.1 Scale Factor Worksheet 2
Goal: To be able to draw a figure with a given scale factor.
Scale factor of 2
Scale factor of
Scale factor: 4
Scale factor:
Scale factor:
Geometry Name
Chapter 11.1 Similar Figures Worksheet 1
1. Given ABCD ~ WXYZ
a. What angles are congruent?
b. Write the proportions that are equal.
2. Given DXYZ~DRST
a. What angles are congruent?
b. Write the proportions that are equal.
3. Explain why the figures are similar and write the similarity statement.
Geometry Name
Chapter 11.1 Similar Figures Worksheet 2
Determine whether the figures are similar. If yes, what is the scale factor that transforms the figure on the left to the figure on the right? Assume the angles are congruent.
1. Similar ? yes no 2. Similar? yes no
If yes, scale factor (left to right) _____ If yes, scale factor (left to right)____
3. Similar ? yes no 4. Similar? yes no
If yes, scale factor (left to right) _____ If yes, scale factor (left to right)____
5. Similar ? yes no 6. Similar? yes no
If yes, scale factor (left to right) _____ If yes, scale factor (left to right)____
Geometry Name
Chapter 11.2 Similar Triangles Worksheet 3
Goal is to understand notation related to similarity and then apply this notation to find a missing side of similar triangles.
Definition of Similar Polygons: Two polygons are similar if and only if the corresponding angles are congruent and the corresponding sides are proportional.
1. These corresponding angles are congruent:
______
______
______
These corresponding sides are proportional:
2.
These corresponding angles are congruent:
______
______
______
These corresponding sides are proportional:
3.
Which angles are congruent? What sides are proportional?
4. What proportions are equal?
Find x Find y
5. What sides are proportional?
x
Find x:
6. Find AC and OG.
Geometry Name
Chapter 11.2 Similar Triangles Worksheet 4
Find the missing lengths of the similar triangles.
1.
Step 1: Write the corresponding sides of and as a proportion:
Step 2: Fill in the numbers and solve for the missing side.
BC = ______
FD = ______
2.
Step 1: Write the corresponding sides of and as a proportion:
Step 2: Fill in the numbers and solve for the missing side.
AC = ______TG = ______
3.
Step 1: Write the corresponding sides of and as a proportion:
Step 2: Fill in the numbers and solve for the missing side.
PM = ______QV = ______
4.
BD = ______EC = ______
Geometry Name
Chapter 11.2 Similar Triangles Worksheet 5
Find the missing lengths. (You may get decimals.)
1. 2.
AC = ______GE = ______RS = ______TR = ______
3.
YZ = ______WY = ______
Similarity Names______
Geometry G
Round Table ______
Find the missing lengths.
1.AC = ______
EG = ______/ 2.
WY = ______
YZ = ______
3.
RS = ______
TR = ______/ 4.
x = ______
NP = ______
QV = ______
Section 9.3 Notes
Methods of proving Triangles Similar
We will look at ways of proving triangles similar.
Recall what similarity means: 1) Corresponding angles are______
2) The ratios of the measures of corresponding sides are______
Postulate: AA to prove triangles similar
Given two corresponding angles congruent, can you prove the triangles similar by AA?
Therefore, AA is a way to prove triangles similar.
The other two ways to prove triangles similar are:
Theorem:
Theorem:
Don’t forget the ~ when proving similar triangles by the three above methods!
*The 3 ways to prove similar triangles are: ______, ______, and ______.
Examples
Decide if each pair of triangles is similar. If they are, write the correspondence in the first blank and the reason in the second blank. If they are NOT similar, write NS in the second blank.
1) ∆ ABC ~ ∆ ______by ______
2) ∆ ABC ~ ∆ ______by ______
3) ∆ YXS ~ ∆ ______by ______
4) ∆ ABC ~ ∆ ______by ______
Geometry Name
Chapter 11.2 Justifying Similar Triangles Worksheet 6
Determine whether each pair of triangles is similar. If the triangles are similar, justify your answer by using SSS, SAS, and AA. Make sure you have work to support your answer.
1.
Yes No ______~ ______by ______
2.
Yes No ______~ ______by ______
3.
Yes No ______~ ______by ______
8.
Yes No ______~ ______by ______
9. Ryan is 5 feet tall. His shadow is 9 feet long and the shadow of a building is 36 feet long. How
tall is the building? Draw two similar triangles and then determine the height of the building.
Geometry Name
Chapter 11.2 Justifying Similar Triangles Worksheet 7
Determine whether each pair of triangles is similar. If the triangles are similar, justify your answer by using SSS~, SAS~, and AA~. Make sure you have work to support your answer.
1.
Yes No
______~ ______
by ______
2. Yes No
______~ ______
by ______
3. Yes No
______~ ______
by ______
4. Yes No
______~ ______
by ______
5. Yes No
______~ ______
by ______
Geometry G Name______
Sec 9.4 Notes
Theorem: If a line is parallel to one side of a triangle and intersects the other two sides, then the triangle formed is similar to the original triangle.
If || , then DABE ~ DACD
Let’s see why this is true.
If || , then the corresponding angles which
are congruent are:
Ð_____ @ Ð______and Ð_____ @ Ð______.
By AA, D ______~ D ______.
Examples
Complete the proportions for the given diagram.
a. b.
c.
We can use these proportions to solve for the missing sides of similar triangles..
1. 2.
Theorem: If a line is parallel to one side of a triangle and intersects the other two sides, then it separates the sides into segments of proportional lengths.
(Also known as the Side-Splitter Theorem.)
If || , then .
If you need to find either BE or CD, you still need
to use similar triangles. You CANNOT use the
Side-Splitter Theorem to find these two sides since
they are not “split” sides.
Examples
Write and solve proportions to solve for each variable.
1. 2.
3. 4.
Geometry
Chapter 11.6 Proportional Segments between Parallel Lines
Directions: Find the value each variable in the diagrams.
1.
2.
3.
Practice 9.4
Solve for x for each problem.
1] 2]
3] 4]
5] 6]
7] 8]
Name______Date______Hour______
Sect 9.5 – Triangle Midsegments
Use centimeters or degrees to find the measures of the following…
SR = ______
RN = ______
SN = ______
SP = ______
PR = ______ÐS = ______ÐN = ______
RI = ______ÐRPI = ______Ð PIR = ______
IN = ______ÐR = ______
PI = ______
Notice anything???
Fill in the measures of all of the sides and angles of the triangle below. Did the same thing occur as above??
THEOREM: The ______of a triangle is ______the length of the third side and is ______to it.
Examples:
1) In the triangle given, A, B, and C are midpoints of the sides of . If TU=12. UV=16 and TV=20…
a) Find AB, BC, and AC
b) Name the three pairs of parallel segments
2) D is the midpoint of and E is the midpoint of .
a. If AD is 8 and AB is 12 find AC, DC, and DE
AC______DC______DE______
b. If , and DE is 17.9 Find and AB
______AB______
c. If and AD is 13 and BC is 27, Find , BE and AC
______BE______AC______
Geometry G Name______
Sec 9.6 Notes Proportional Parts and Parallel Lines
Remember the Side-Splitter Theorem?
Theorem: If a line is parallel to one side of a triangle and intersects the other
two sides, it divides those two sides proportionally.
Given: ||
Prove:
What happens if there are more than two parallel lines?
Theorem: If three or more parallel lines intersect two transversals, the parallel lines divide the transversals proportionally.
Given: || ||
Conclusion:
Examples:
1. Complete each proportion.
a.
b.
c.
Write and solve a proportion to find the value of x.
2.
3.
4.
5.
Name______Date______Hour______
Sect. 9.7 – Perimeters and Similarity
1) Use the Pythagorean Theorem to find AC and DE.
AC = ______
DE = ______
2) Find the following ratios.