7.1: Ratios and Proportions
Ratio:
Ratios are written in 3 forms:
Ratios must always be reduced to simplest form.
Extended Ratios: when a ratio contains 3 or more numbers, a:b:c (sides of a triangle)
When writing ratios, ORDER does matter as well as units. READ what is asked.
Examples: Write the ratio of the first measurement to the second
measurement.
1. diameter of a salad plate: 8 in.
3. garden container width: 24 in.
diameter of a dinner plate: 12 in.
garden container length: 8 ft
2. weight of a cupcake: 2 oz
weight of a cake: 24 oz
4. height of a book: 11 in.
height of a bookshelf: 3 ft 3 in.
5. The perimeter of a triangle is 270 cm. The ratio of the sides is 2 : 3 : 4. What is the length of
each side?
Proportions:
Proportions are solved using cross products.
Algebra Solve each proportion.
Examples: Solve each proportion.
6.
x 13

4 52
7.
x
Examples: Use the proportion z
9.
x

6
10.
xz

z

x
16

2 x  1 40
8.
2 b 1

7 56
6
5 . Complete each statement. Justify your answer.
11.
z

x
12. 5x 
7.2: Similar Polygons
Similar Figures:
figures that have the same general shape, but size might not be the same
Similar Polygons:
Scale Factor:
The scale factor can be written as an extended proportion.
Example: List the pairs of congruent angles and the extended proportion that relates
the corresponding sides for the similar polygons.
1. ABCD ~ WXYZ
Determining If Polygons are Similar
1) Are all corresponding angles congruent?
2) Do all corresponding sides have the same scale factor?
Examples: Determine whether the polygons are similar. If so, write a similarity statement and give
the scale factor. If not, explain.
2.
3.
Using Similarity to Solve Problems
Angles in similar polygons are
.
In order to find values associated with angles, set things equal.
Corresponding Sides in similar polygons are
In order to find values associated with sides, set up a proportion.
.
Example:
4) Find the value of y. Give the scale factor of the polygons.
ABCD ~ TSVU
Examples: In the diagram below, PRQ ~ DEF. Find each of the following.
5. the scale factor of PRQ to DEF
6. mD
7. mR
8. mP
9. DE
10. FE
7.3: Proving Triangles Similar
What must be true about similar polygons? (2 things)
Diagrams for AA~, SAS~ and SSS~
Examples: Determine whether the triangles are similar. If so, write a similarity statement and name the
postulate or theorem you used. If not, explain.
1.
2.
3.
4.
5.
6.
Examples: Explain why the triangles are similar. Then find the value of x.
7.
8.
9.
x+5
24
4
10. A stick 2 m long is placed vertically at point A. The top of the stick is in line with the top of a
tree as seen from point A, which is 3 m from the stick and 30 m from the tree. How tall is the
tree? Let the tree be x.
7.4: Similarity in Right Triangles
Examples: Find the geometric mean between the given numbers. Simplify all radicals.
1. 4 and 9
2) 4 and 18
3) 6 and 15
The “T” Method
Used when you are given the altitude inside.
Examples: Determine the value of x I the given triangles.
4)
5)
The “Arrow” Method
Used when legs and hypotenuses are given.
Nothing on the inside usually
Examples: Find the value of x.
6)
7)
Examples: Find the value of each variable in the following.
8)
9)
10)
7.5: Proportions in Triangles
When parallel lines are formed inside triangles, proportions follow.
Side-Splitter Theorem: If a line is parallel to one side of a triangle and intersects the other two
sides, then it divides those sides proportionally.
Examples: Solve the following for x.
1.
2.
3.
Corollary to the Side-Splitter Theorem: If 3 parallel lines intersect two transversals, then the
segments intercepted on the transversals are proportional.
Examples: Find the value of x in the following.
4.
5.
Examples: Find the value of x in the following.
6.
7.