6.1 Ratios, Proportions, and the Geometric Mean
Obj.: Solve problems by writing and solving proportions.
Key Vocabulary
• Ratio - If a and b are two numbers or quantities and b≠ 0, then the ratio of a to b
is a/b. The ratio of a to b can also be written as a: b.
• Proportion - An equation that states that two ratios are equal is called a proportion.
• Means, extremes - The numbers b and c are the means of the proportion. The
numbers a and d are the extremes of the proportion.
♦♦♦♦♦KEY CONCEPT♦♦♦♦♦
A Property of Proportions
1. Cross Products Property In a proportion, the product of the extremes
equals the product of the means.
If a = c where b ≠ 0 and d ≠ 0, then a • d = b • c.
b
d
4
2
= 6,
3
2 • 6 = 3 • 4,
12 = 12
• Geometric mean - The geometric mean of two positive numbers a and b is the
x
a
positive number x that satisfies x = b . So, x² = ab and x = ab .
EXAMPLE 1 Simplify ratios
Simplify the ratio. (See Table of Measures, p. 921)
a. 76 cm: 8 cm
b.
4 ft
24 in.
Solution
EXAMPLE 2 Use a ratio to find a dimension
PAINTING You are painting barn doors. You know that the perimeter of the wall is 64 feet and
that the ratio of its length to its height is 3:5. Find the area of the wall.
Solution
EXAMPLE 3 Use extended ratios
ALGEBRA The measures of the angles in ▲BCD are in the extended ratio of 2: 3: 4. Find the
measures of the angles.
Solution
EXAMPLE 4 Solve proportions
ALGEBRA Solve the proportion.
3
x
a.
=
4
16
Solution
b.
3
2
=
x 1
x
EXAMPLE 5 Solve a real-world problem
Bowling You want to find the total number of rows of boards that make up 24 lanes at a
bowling alley. You know that there are 117 rows in 3 lanes. Find the total number of rows of
boards that make up the 24 lanes.
Solution
EXAMPLE 6 Find a geometric mean
Find the geometric mean of 16 and 48.
Solution
6.1 Cont.
6.2 Use Proportions to Solve Geometry Problems
Obj.: Use proportions to solve geometry problems.
Key Vocabulary
• Scale drawing - A scale drawing is a drawing that is the same shape as the
object it represents.
• Scale - The scale is a ratio that describes how the dimensions in
the drawing are related to the actual dimensions of the object.
♦♦♦♦♦KEY CONCEPT♦♦♦♦♦
Additional Properties of Proportions
2. Reciprocal Property If two ratios are
equal, then their reciprocals are also
equal.
If
a
b
=
b
c
, then
a
d
=
d
c
3. If you interchange the means of a
proportion, then you form another
a
c
a
b
If b = d , then c = d
true proportion.
4. In a proportion, if you add the value
of each ratio’s denominator to its
numerator, then you form another
true proportion.
If
a
b
EXAMPLE 1 Use properties of proportions
AC
In the diagram,
= BC . Write four true proportions.
DF
EF
Solution
EXAMPLE 2 Use proportions with geometric figures
JL
ALGEBRA In the diagram,
= JK . Find JH and JL.
LH
Solution
KG
=
c
a  b
c d
, then
=
d
b
d
EXAMPLE 3 Find the scale of a drawing
Keys The length of the scale drawing is 7 centimeters. The length of the
actual key is 4 centimeters. What is the scale of the drawing?
Solution
EXAMPLE 4 Use a scale drawing
MAPS The scale of the map at the right is 1 inch: 8 miles.
Find the actual distance from Westbrook to Cooley.
Solution
EXAMPLE 5 TAKS Reasoning: Multi-Step Problem
SCALE MODEL You buy a 3-D scale model of the Sunsphere in Knoxville,
TN. The actual building is 266 feet tall. Your model is 20 inches tall, and the
diameter of the dome on your scale model is about 5.6 inches.
a. What is the diameter of the actual dome?
b. About how many times as tall as your model is the
actual building?
Solution
6.2 Cont.
6.3 Use Similar Polygons
Obj.: Use proportions to identify similar polygons.
Key Vocabulary
• Similar polygons - Two polygons are similar polygons if corresponding angles
are congruent and corresponding side lengths are proportional.
• Scale factor - If two polygons are similar, then the ratio of the lengths of
two corresponding sides is called the scale factor.
Corresponding angles
∠A ≅ ∠E, ∠B ≅ ∠F, ∠C ≅ ∠G,
and ∠D ≅ ∠H
ABCD ∼ EFGH
Ratios of corresponding sides
AB
EF
Order Matters!!
=
BC
FG
=
CD
GH
=
DA
HE
Perimeters of Similar Polygons Theorem
If two polygons are similar, then the
ratio of their perimeters is equal to
the ratios of their corresponding
side lengths.
If KLMN ∼ PQRS, then
KL  LM  MN  NK
PQ  QR  RS  SP
=
KL
PQ
=
LM
QR
=
MN
RS
=
NK
SP
.
♦♦♦♦♦KEY CONCEPT♦♦♦♦♦
Corresponding Lengths in Similar Polygons
If two polygons are similar, then the ratio of any two corresponding
lengths in the polygons is equal to the scale factor of the similar polygons.
EXAMPLE 1 Use similarity statements
In the diagram, ▲ABC ∼ ▲DEF.
a. List all pairs of congruent angles.
b. Check that the ratios of corresponding side lengths are equal.
c. Write the ratios of the corresponding side lengths in a statement of proportionality.
Solution
EXAMPLE 2 Find the scale factor
(6.3 cont.)
Determine whether the polygons are similar. If they are, write a similarity
statement and find the scale factor of ABCD to JKLM.
Solution
EXAMPLE 3 Use similar polygons
ALGEBRA In the diagram, ▲BCD ∼ ▲RST. Find the value of x.
Solution
EXAMPLE 4 Find perimeters of similar figures
Basketball A large cement court is being poured for a basketball hoop in place of a smaller
one. The court will be 20 feet wide and 25 feet long. The old court was similar in shape, but
only 16 feet wide.
a. Find the scale factor of the new court to the old court.
b. Find the perimeter of the new court and the old court.
Solution
EXAMPLE 5 Use a scale factor
In the diagram, ▲TPR ∼ ▲XPZ. Find the length of the altitude GL.
Solution
6.3 Cont.
6.4 Prove Triangles Similar by AA
Obj.: Use the AA Similarity Postulate.
Key Vocabulary
• Similar polygons - Two polygons are similar polygons if corresponding angles are
congruent and corresponding side lengths are proportional.
Angle-Angle (AA) Similarity Postulate
If two angles of one triangle are congruent
to two angles of another triangle, then the
two triangles are similar.
EXAMPLE 1 Use the AA Similarity Postulate
Determine whether the triangles are similar. If they are, write
a similarity statement. Explain your reasoning.
Solution
EXAMPLE 2 Show that triangles are similar
Show that the two triangles are similar.
a. ∆RTV and ∆RQS
Solution
b. ∆LMN and ∆NOP
∆JKL ~ ∆XYZ
EXAMPLE 3 Using similar triangles
Height A lifeguard is standing beside the lifeguard chair on a beach. The lifeguard is 6 feet 4
inches tall and casts a shadow that is 48 inches long. The chair casts a shadow that is 6 feet
long. How tall is the chair?
Solution
6.5 Prove Triangles Similar by SSS and SAS
Obj.: Use the SSS and SAS Similarity Theorems.
Key Vocabulary
• ratio, p. 356
• proportion, p. 358
• similar polygons,
p. 372
Side-Side-Side (SSS) Similarity Theorem
If the corresponding side lengths of
two triangles are proportional, then
the triangles are similar.
If
= =
, then ∆ABC ~ ∆RST.
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.
If ∠X ≅ ∠M and
=
, then ∆XYZ ~ ∆MNP
EXAMPLE 1 Use the SSS Similarity Theorem
Is either ∆DEF or ∆GHJ similar to ∆ABC?
Solution
EXAMPLE 2 Use the SSS Similarity Theorem
ALGEBRA Find the value of x that makes ∆ABC ~∆DEF.
Solution
6.6 Use Proportionality Theorems
Obj.: Use proportions with a triangle or parallel lines.
Key Vocabulary
• Corresponding angles - Two angles are corresponding
angles if they have corresponding positions. For example,
∠2 and∠6 are above the lines and to the right of the transversal t.
• Ratio - If a and b are two numbers or quantities and b ≠ 0, then the ratio of a to b
is a/b. The ratio of a to b can also be written as a : b.
• Proportion - An equation that states that two ratios are equal is called a proportion.
Triangle Proportionality Theorem
If a line parallel to one side of a triangle
intersects the other two sides, then it
divides the two sides proportionally.
∆ prop. Th
Converse of the Triangle Proportionality Theorem
If a line divides two sides of a triangle
proportionally, then it is parallel to the
third side.
If ̅̅̅̅⫽ ̅̅̅̅, then
If
=
, then ̅̅̅̅⫽ ̅̅̅̅
3 ⫽ lines inters. 2 trans. → prop.
If three parallel lines intersect two
transversals, then they divide the
transversals proportionally.
=
ray bis. ∠ of ∆→opp. side prop. to other 2 sides
If a ray bisects an angle of a triangle, then
it divides the opposite side into segments
whose lengths are proportional to the
lengths of the other two sides.
EXAMPLE 1 Find the length of a segment
In the diagram,̅̅̅̅⫽ ̅̅̅̅ , RQ = 10, RS = 12, and ST = 6,
What is the length of ̅̅̅̅?
Solution
=
=
EXAMPLE 2 Solve a real-world problem
Aerodynamics A spoiler for a remote controlled car is shown where AB = 31 mm, BC = 19,
CD = 27 mm, and DE = 23 mm. Explain why ̅̅̅̅ is not parallel to ̅̅̅̅.
Solution
EXAMPLE 3 Use Theorem 6.6
Farming A farmer’s land is divided by a newly constructed interstate.
The distances shown are in meters. Find the distance CA between
the north border and the south border of the farmer’s land.
Solution
EXAMPLE 4 Use Theorem 6.7
In the diagram, ∠ DEG ≅ ∠ GEF. Use the given
side lengths to find the length of ̅̅̅̅ .
Solution
6.6 Cont.