Download Triangles - pcrest3.com

Section 3.2
Triangles
Pre-Activity
Preparation
Geena needs to make sure that the deck she is building is perfectly square to the brace
holding the deck in place. How can she use geometry to ensure that the boards are
aligned properly?
Geena can use the Pythagorean Theorem (see information below). Every right triangle
has special properties related to the lengths of its sides, its angles, and even the
relationship of its angles to its sides. The mathematical field
of trigonometry is the study of right triangle relationships.
Each deck board must make a square corner with the brace. A
square corner is a right angle. Geena can use the Pythagorean
Theorem to ensure that the corner is a right angle by measuring the two sides of a
triangle formed by the brace and board. She will measure 4″ along one side and make
a point, and 3″ along the other side and make a point. If the line drawn connecting the
two points measures exactly 5″, then the angle opposite that line is a right angle.
5”
4”
3”
Learning Objectives
• Learn how to apply the Pythagorean Theorem to find the length of a side of a right triangle, given its
other two sides
• Use the rules for congruent triangles to determine if two triangles are congruent
• Find the perimeter and area of a triangle given its base and height
Terminology
Previously Used
New Terms
to
Learn
acute angle
area
isosceles
angle
base
legs
obtuse angle
congruent triangles
perimeter
perpendicular
corresponding sides
Pythagorean Theorem
square root
equilateral
right triangle
vertex
height
scalene
hypotenuse
side
173
Chapter 3 — Geometry
174
Building Mathematical Language
Triangles
Three distinct non-parallel lines, lying in the same plane, will intersect at three
distinct points and form a triangle. Triangles are named based upon these points
of intersection. The triangle to the right is called “triangle ABC” or “ABC.”
(While triangle names are usually alphabetized—“ABC” rather than “CAB”—
this is not always the case.)
a
B
b
A
c
C
• The interior angles of the triangle are a, b, and c. The sum of the
interior angles of a triangle is always 180°: ∠a + ∠b + ∠c = 180°
• The base or bottom of triangle ABC is the segment BC, AB, or AC depending on the orientation of
the triangle. (Orientation means the way the triangle is turned).
• The height (also called the altitude) is the perpendicular distance from the top point to the base.
Find the height by measuring the length of a line drawn from the top vertex that intersects the base
at a right angle.
• Each pair of intersecting lines forms four angles—two pairs of vertical angles.
• Each interior angle is adjacent to two supplementary exterior angles and across from one vertical
exterior angle.
Types of Triangles
Name
Picture
Special Features
Observations
No sides are equal and no
angles are equal
Scalene triangles are the
“generic” triangle and
have no special features.
Two equal sides and two
equal angles
Usually presented with
the base and equal angles
at the bottom. The equal
sides are opposite the
equal angles.
Equilateral
All angles are equal and all
sides are equal
An equilateral triangle has
angles that measure 60°.
Acute
All three angles are acute
Each angle measures less
than 90°. An equilateral
triangle is acute.
One angle is obtuse
One obtuse angle;
therefore, two acute
angles.
Scalene
Isosceles
Obtuse
Section 3.2 — Triangles
175
Name
Picture
Special Features
Observations
Right
One angle is a right angle.
The side opposite the right
angle is the hypotenuse.
The other two sides can be
called legs.
Many concepts in
mathematics and many
real world applications use
right triangles.
Isosceles
Right
The legs of the right angle
are equal and the two
acute angles are equal and
measure 45°.
Dividing a square on one
of its diagonals creates two
isosceles right triangles.
Dividing a square on both
its diagonals yields four
isoceles right triangles.
Perimeter of a Triangle
Perimeter is the distance around an object in a plane. Perimeter is a measure of length and, therefore,
measured in feet (ft), inches (in), meters (m), centimeters (cm), etc. In this context, the Greek peri- means
around and -meter means measure, so the word “perimeter” means measure around.
The perimeter of a triangle is the sum of the measures of the sides:
Perimeter
P=a+b+c
B
The perimeter of triangle ABC
P=a+b+c
P = 4.9 + 7.8 + 12
P = 24.7
c = 12
A
b = 7.8
a = 4.9
C
Area of a Triangle
Area measures the amount of surface of an object in a plane. Measure area in terms of square feet (ft2),
square meters (m2), square miles (mi2), etc.
The formula for finding the area of a triangle is one-half times the base times the height:
Area
A = ½bh
B
Area = ½ base × height
The area of right triangle ABC
A = ½bh
A = (½)(12m)(4m)
A = (6 × 4)m2
A = 24m2
height
= 4m
Validate: 24m2 ÷ 4m = 6m; 6m ÷ 12m = ½
C base (b) = 12m
A
Chapter 3 — Geometry
176
Congruent Triangles
Triangles that are the same size and shape are congruent triangles. Congruent triangles have equal
corresponding angles and equal corresponding side lengths. Use the symbol “,” to indicate
congruency.
A
b=7
C
a=5
D
e=7
c = 3.5
B
F
f = 3.5
d=5
E
∠A , ∠D
and
a,d
∠B , ∠E and
b,e
∠C , ∠F
and
c,f
Triangle ABC , DEF
Techniques
Determining Congruency between Triangles
To determine if two triangles are congruent, match their corresponding parts and then use
one of the following rules.
Types of Triangle Congruency
Name
S-S-S
side–side–side
S-A-S
side-angle-side
Description
Two triangles
are congruent if
corresponding sides
are equal.
Two sides and the
enclosed angle of one
triangle are equal to
the corresponding
two sides and
enclosed angle of the
other triangle.
Two angles and the
enclosed side of one
triangle are equal to
the corresponding
angle-side-angle two angles and
enclosed side of the
other triangle.
Visual
The tick marks indicate which
sides are conguent.
Observations
If you can determine
that corresponding
side measures
are equal, then
the triangles are
congruent.
The equal angle must
be between the two
equal sides.
The tick marks indicate
conguent sides and angles.
A-S-A
The tick marks indicate
conguent angles and sides.
The equal side must
be between the two
equal angles.
Section 3.2 — Triangles
177
Pythagorean Theorem
The Pythagorean Theorem is the most widely used and important relationship in geometry.
Knowledge of this theory has existed for millennia, with the first recorded statement found on a Babylonian
tablet (c. 1600 B.C.). While various sources contain proofs of the relationship (some of which certainly
predate Pythagoras’ work), the theorem was named after him, as it was his work that became most widely
known.
Pythagorean Theorem
In any right triangle ABC, where c is
the side opposite the right angle:
B
c
a
a2 + b2 = c2
(Note: side a is across from angle A
and side b is across from angle B.)
C
b
A
Example: Given right triangle ABC, and a = 3, b = 4, and c = 5:
a2 + b2 = c2
32 + 42 = 52
9 + 16 = 25
25 = 25
B
3
C
5
4
A
Write down the Pythagorean Theorem for the triangle below. Label the right angle as
angle C and c = 10, b = 8, and a = 6.
Try it!
a2 + b2 = c2
B
a
2
a
C
Area of Square a + Area of Square b = Area of Square c
c2
You can easily learn more about the Pythagorean Theorem online.
c
Here are some fun sites to get you started:
A
b
http://www.cut-the-knot.org/pythagoras/
http://www.mathsnet.net/dynamic/pythagoras/
b
2
http://mathworld.wolfram.com/PythagoreanTheorem.html
http://www.pbs.org/wgbh/nova/proof/puzzle/theorem.html
a2 + b2 = c2
Chapter 3 — Geometry
178
Methodology
Using the Pythagorean Theorem: a2 + b2 = c2
Label each triangle with
C as its right angle.
►
Example 1: Find c if a = 7 and b = 12.
Round to the nearest tenth, if necessary.
C
►
Example 2: Find c if a = 11 and b = 15.
Round to the nearest tenth, if necessary.
Steps in the Methodology
Step 1
Identify a,
b and c on
the given
triangle.
Any of the three sides can be
the unknown value.
Make a sketch if necessary.
Substitute
the given side
lengths into
the formula.
Example 1
c=?
a=7
C
b = 12
a=7
b =12
c = unknown
C
Step 2
Try It!
If c is not the unknown
side, use the equivalent
formula to find side a or b:
a2 + b2 = c2
72 + 12 2 = c2
c2 – b2 = a2
or
c – a2 = b2
2
Step 3
Solve for the
square of the
unknown side
using order of
operations.
Step 4
Use your
calculator
to find the
positive
square root.
Step 5
Validate
Solve for a2 or b2 if you
are using the alternative
formulas. Use your calculator
if necessary.
49 + 144 = c2
Round off to the desired
number of places. If you
round off, use ≈ not =.
193 = c 2
Use the original information
to check for equality.
72 + 12 2 = 13.9 2
193 = c2
193 = c
13.9 . c
?
?
49 + 144 = 193.2
193 ≈ 193.2
Example 2
Section 3.2 — Triangles
179
Models
B
Model 1
Find the perimeter and area of the given right triangle.
c = 24.5 in
a = h = 14.7 in
C
Perimeter
P=a+b+c
A
b = 19.6 in
Validate: 58.8 in – 24.5 in = 34.3 in
P = 14.7 + 19.6 + 24.5
34.3 in – 19.6 in = 14.7 in 
P = 58.8 inches
Area
A = ½bh
Validate:
A = ½(19.6)(14.7)
A = (9.8)(14.7)
A = 144.06 square inches
(144.06 in 2 )
= 9.8 in
14.7 in
(9.8 in )
= (9.8 in ) × 2 = 19.6 in 
1
2
B
Model 2
Find the perimeter and area of the given triangle.
a = 50 in
c = 41 in
h = 40 in
C
Perimeter
determine side b (base):
30 in
9 in
A
Validate: 130 in – 41 in = 89 in
30 in + 9 in = 39 in
89 in – 50 in = 39 in 
P=a+b+c
P = 50 + 39 + 41
P = 130 inches
Area
A = ½bh
A = ½(39)(40)
A = (19.5)(40)
A = 780 square inches
Validate:
(780 in 2 )
= 19.5 in
40 in
(19.5 in )
= (19.5 in ) × 2 = 39 in 
1
2
Chapter 3 — Geometry
180
Model 3
Given a right triangle, find side b if side a = 3.9 and side c (hypotenuse) = 6.5. Approximate the answer to
the nearest tenth, if necessary.
Step 1
Identify
a = 3.9
c = 6.5
a = 3.9
b = unknown
b
c = 6.5
Step 2
Substitute
c2 – a2 = b2 (Alternative formula to find side b.)
6.52 – 3.92 = b2
Step 3
Solve
42.25 – 15.21 = b2
27.04 = b2
Step 4
Square root
5.2 = b
Step 5
Validate
5.22 + 3.92 = 6.52
?
?
27.04 + 15.21 = 42.25
42.25 = 42.25 
Model 4
Given a right triangle, find side a if side b = 11 and side c (hypotenuse) = 18.2.
Approximate the answer to the nearest tenth, if necessary.
a
Step 1
Identify
c = 18.2
a = unknown
b = 11
c = 18.2
Step 2
Substitute
c2 – b2 = a2
b = 11
18.22 – 112 = a2
Step 3
Solve
331.24 – 121 = a2
210.24 = a2
Step 4
Square root
14.5 ≈ a
Step 5
Validate
14.52 + 112 = 18.22
?
?
210.25 + 121 = 331.24
331.25 ≈ 331.24

Section 3.2 — Triangles
181
Model 5a
B
Are the triangles in the pair at right congruent? Use what
you know about triangles, including types of congruency
and how to solve for unknown sides in right triangles, in
order to determine if the triangles at right are congruent with
each other. Provide an explanation for your answer(s).
E
45˚
a
45˚
1
C
c
d = 3m
b = 3m
A
F
The missing angle in triangle 1 (∠A) is 45°: ∠A = 180° – 90° – 45° = 45°
Triangle 2 is also an isosceles triangle and side e = 3m.
With the enclosed angle of 90°, the S-A-S rule of congruity is satisfied. See the
diagram below.
E
45˚
a = 3m
C = 90°
ANGLE
SIDE 1
45˚
b = 3m
A
ANGLE
45˚
e = 3m
D
SIDE 2
We can also apply the Pythagorean Theorem to find the length of the hypotenuse, to
show S-S-S congruency.
a = 3m, b = 3m
c2 = 32 + 32 = 18
c ≈ 4.24 m
d = 3m, e = 3m
f 2 = 32 + 32 = 18
f ≈ 4.24 m
B
E
45˚
a = 3m
SIDE 1
C
45˚
c = 4.24m
SIDE 3
d = 3m
1
45˚
b = 3m
SIDE 1
A
SIDE 2
F
f = 4.24m
2
SIDE 3
45˚
e = 3m
A-S-A
Angle 1: ∠C , ∠F
Side:
b,e
Angle 2: ∠A , ∠D
Knowing the missing hypotenuse for each triangle
also enables us to show S-A-S congruency in two
additional ways:
S-S-S
Side 1: a , d
Side 2: b , e
Side 3: c , f
1 , 2
D
SIDE 2
Once we have used the Pythagorean Theorem to find sides c and f, we have all angle
and side measurements for each triangle. This allows us to show A-S-A congruency,
in three different ways:
A-S-A
Angle 1: ∠B , ∠E
Side:
a,d
Angle 2: ∠C , ∠F
1 , 2
2
F = 90°
SIDE 2
Side 1: a , d
Angle: ∠C , ∠F
Side 2: b , e
45˚
d = 3m
1
45˚
e
S-A-S
Triangle 1 is therefore an isosceles triangle and side a = 3m.
SIDE 1
2
Type of
Congruency
Reason
B
f
A-S-A
1 , 2
A-S-A
Angle 1: ∠A , ∠D
Side:
c,f
Angle 2: ∠B , ∠E
S-A-S
Side 1: a , d
Angle: ∠B , ∠E
Side 2: c , f
S-A-S
Side 1: c , f
Angle: ∠A , ∠D
Side 2: b , e
D
Chapter 3 — Geometry
182
Model 5b
Are the triangles in the pair at right congruent? Use what you know about
triangles, including types of congruency and how to solve for unknown
sides in right triangles. Provide an explanation for your answer.
1.8
48˚
80˚
52˚
1
2
x
52˚
1.8
2.3
Type of
Congruency
Reason
These are not right triangles, so we cannot use the Pythagorean Theorem to find
missing side lengths. However, we already have equivalent sides of 1.8 and
equivalent angles of 52°. If we can solve for the missing angle (∠x) in triangle 2
and it is equal to 80°, then we can show A-S-A congruency.
∠x = 180° – 52° – 48° = 80°. The 80° angles, the 52° angles, and their enclosed
sides of 1.8 satisfy the A-S-A rule of congruency. See the diagram below.
A-S-A
Angle 1: 80° = 80°
Side:
1.8 = 1.8
Angle 2: 52° = 52°
1 , 2
ANGLE 1
SIDE
1.8
ANGLE 2
80˚
48˚
52˚ 1
2.3
2
ANGLE 1
80˚ 52˚
1.8
SIDE
ANGLE 2
Model 5c
Determine whether or not the triangles in the
pair at right congruent. Use what you know
about triangles, including types of congruency
and how to solve for unknown sides in right
triangles. Provide an explanation for your
answer.
12.3
99°
8.7
99°
12.3
21.6
Reason
Keep in mind that you cannot determine congruency by “looks.” You must use
S-S-S, S-A-S, or A-S-A to show congruency. The largest corresponding angles
equal 99°, so they are not right triangles and the Pythagorean Theorem cannot be
used. While the triangle on the right has the given angle enclosed with the two given
sides, the triangle on the left does not; therefore, the relationship does not fit any of
the three rules for congruency.
Type of
Congruency
NOT
CONGRUENT
Section 3.2 — Triangles
183
Addressing Common Errors
Incorrect
Process
Issue
Misidentifying
corresponding
sides or angles
If triangle ABC
is congruent to
triangle XYZ,
Orientation
matters. Rearrange
the triangles so that
the largest angles
are matched.
(9 ABC / 9XYZ)
which sides are
equal measures?
c
a
C
z
A
b
C
Z
y
X
Y
c
a
x
b
If side a = 12 and
side b = 6, what is
the measure of side
c?
a
z
x
A
Z
y
X
a=x
b=y
c=z
a=z
c=x
b=y
Use of the
Pythagorean
theorem on
non-right
triangles
Validation
The second triangle
should be rotated
or flipped until
angle Z is in the
same position as
angle C.
B
Y
B
Correct
Process
Resolution
The Pythagorean
Theorem is only
applicable to RIGHT
triangles.
b
c
c 2 = a 2 + b 2 , so
Careful observation
reveals that the
triangle is not a
right triangle; no
indication of a right
angle is present.
The Pythagorean
Theorem is not
useful here.
Side c cannot be
determined with the
information given.
c 2 = 122 + 62 and
c 2 = 144 + 36
c 2 = 180

c = 180 . 13.4
Applying the
Pythagorean
Theorem
incorrectly
In right
triangle
ABC:
A
c
b
C
a
if a = 5 and b = 7
find c.
a+b=c
5 + 7 = 12 = c
B
The Pythagorean
Theorem states that
the SQUARES of
the two legs added
together equals
the SQUARE of the
hypotenuse.
?
a2 + b2 = c2
52 + 72 = 8.62
52 + 72 = c2
25 + 49 = 73.96
25 + 49 = c2
74 = c2
c = 74 . 8.6
?
74 ≈ 73.96


Chapter 3 — Geometry
184
Incorrect
Process
Issue
Using the
wrong
side as the
hypotenuse
in the
Pythagorean
formula
Triangle RST is a
right triangle. Find
side s if r = 3 and
t=5
The hypotenuse is
always opposite the
right angle.
T
s
r
S
Correct
Process
Resolution
Validation
In the given figure,
the right angle is
opposite side t, not
side s. Use is the
alternative formula
for finding side s:
s2 = t2 – r2.
R
t
t2 = r2 + s2
?
52 = 32 + 42
?
25 = 9 + 16
25 = 25 
s2 = t2 – r2
s2 = r 2 + t 2
s2 = 52 – 32
s 2 = 9 + 25
s2 = 16
s2 = 25 – 9
s 2 = 32 + 52
s=4
s 2 = 34
s = 34
s . 5.8
Not using
square units
for area
Find the area of a
triangle whose base
is 10 mm and height
is 6mm.
A = ½bh
A = (½)(6)(10)
Area is always
given in square
units. Carry the
units along with the
measures to remind
you to report the
units as part of the
answer.
A = ½bh
(mm)(mm) = mm2 
A = (½)(6mm)(10mm)
A = 30 mm2
A = 30 mm
Not using an
appropriate
piece of
information in
a formula.
Find the area of the
given triangle:
A
c
B
b
a
C
When a = 6,
b = 10, and h = 8.
A = ½bh
A = (½)(6)(10)
A = 30 square units
Make sure that
you correctly
identify each piece
of information
before using it in
a formula. Clearly
labeling a drawing
or sketch can help.
A
b = 10
c
h=8
B
a=6
C
From the new
drawing, the height
is clearly shown.
A = ½bh
A = (½)(6)(8)
A = 24 square units
Using the area
calculated, solve
for h:
A = ½bh
24 = (½)(6)(h)
24 = 3h
24
=h
3
h = 8, the height
as given. 
Section 3.2 — Triangles
185
Preparation Inventory
Before proceeding, you should be able to:
Find the area and perimeter of a right triangle
Find the perimeter and area of a triangle, given its base and height.
Use the Pythagorean Theorem to find the unknown side of a right triangle
Find the area and perimeter of triangles using the Pythagorean Theorem
Use the congruency rules to establish the congruency of two triangles
CAN a Triangle Have Three 90-Degree Angles?
Of course not, but it is a fun optical illusion. It was first created by
the Swedish artist Oscar Reutersvärd in 1934, but later independently
devised and made popular by the mathematician Roger Penrose. He
described it as “impossibility in its purest form.” You can learn more by
searching the internet for the “Penrose triangle.”
Impossible Triangle sculpture by Brian MacKay
& Ahmad Abas. It is located at Claisebrook
roundabout, East Perth, Western Australia.
This Penrose triangle only appears to have
three 90-degree angles when seen from
certain perspectives.
Section 3.2
Activity
Triangles
Performance Criteria
• Finding the third side of a right triangle when given the measures of two of its sides
– appropriate use of the Pythagorean Theorem to find the third side
– correct calculation of the measure of the third side
• Calculating perimeters and areas of triangles
– correct and appropriate use of the perimeter formula for a triangle
– correct and appropriate use of the area formula for a triangle
• Calculating the measures of sides and angles in two given triangles
– correct identification of corresponding parts
– correct calculation of missing angle or side measures
– correct determination of congruency
Critical Thinking Questions
1. If two triangles are congruent by the S-S-S rule, are the triangles always equilateral triangles? Explain.
2. If you know that two triangles have two pairs of equal corresponding angles, what can you determine
about the third pair of angles?
3. Why can’t “angle-angle-angle” be a rule to determine two congruent triangles?
186
Section 3.2 — Triangles
187
4. Can a triangle have two obtuse angles? Explain your answer.
5. Why can you eliminate the negative square root when finding the square root in the Pythagorean
Theorem?
6. Why is the height needed to measure the area of a triangle?
7. Given a triangle with the following side lengths: side a = 3 feet side b = 4 meters side c = 2 yards,
can you calculate the perimeter? If so, what units would your answer have? Explain your reasoning.
Chapter 3 — Geometry
188
Tips
for
Success
• If a figure is not provided, make a quick sketch showing all measurements
• Use graph paper to sketch more accurately
• Make sure the measurements are in the same units
Demonstrate Your Understanding
1. Find the perimeter of the following triangles:
Problem
Worked Solution
a)
22.5 in
18 in
13.5 in
b)
18.5 m
6.8 m
6m
20.7 m
c)
12 m
15 m
9m
d)
17.3 cm
10.8 cm
9.6 cm
Validation
Section 3.2 — Triangles
189
2. Find the area of the following triangles:
Problem
Worked Solution
a)
18 in
22.5 in
13.5 in
b)
18.5 m
6m
6.8 m
20.7 m
c)
A right triangular
tract of land has
side a = 41.6 miles
and side b = 31.2
miles.
31.2 mi
41.6 mi
d)
11 in
12 in
Validation
Chapter 3 — Geometry
190
3. Use the Pythagorean Theorem to solve the following problems. Round your answers to the nearest tenth,
if necessary.
Problem
a)
Worked Solution
Find c.
c
11 in
12 in
b)
Find b.
20 in
15 in
b
c)
Find a.
a
c = 19.4
b = 13
d)
Find c and e.
c=?
a = 7.5
h=4
e=?
d = 4.2
Validation
Section 3.2 — Triangles
191
Problem
Worked Solution
e)
Find the perimeter of
the largest triangle in
problem 3 d).
f)
Find the area of the
largest triangle in
problem 3 d).
Validation
4. Are the triangles in the following pairs congruent? Use what you know about triangles, including types of
congruency and how to solve for unknown sides in right triangles, in order to determine if the triangles are
congruent with each other. Provide an explanation for your answer(s).
Triangle Pair
Reason
a)
3.9 in
1
c
5.2 in
6.5 in
2
d
5.2 in
Type(s) of
Congruency
Chapter 3 — Geometry
192
Triangle Pair
Reason
b)
7 cm
A
60˚ 7 cm
1
7 cm
c)
B
27˚
e
60˚ 7 cm
60˚
2
f
x
33˚
w
120˚
27˚
E
Type(s) of
Congruency
Section 3.2 — Triangles
Identify
and
193
Correct
the
Errors
In the second column, identify the error(s) in the worked solution or validate its answer.
If the worked solution is incorrect, solve the problem correctly in the third column and validate your answer.
Identify Errors
or Validate
Worked Solution
1) The two right triangles
below are congruent. If
∠1 = 43°, what is the
measure of ∠2?
2
1
∠1 = ∠2
(corresponding parts)
so ∠2 = ∠43°
2) Find the measure of side b
if c = 22 in and a = 8 in.
B
A
C
c2 = b2 + a2
82 = b2 + 222
64 = b2 + 484
b2 = 420
b ≈ 20.5 in
3) Find the measure of side a
if c = 2.5 in and b = 1.5 in.
B
C
A
a2 = c2 – b2
a2 = 2.52 – 1.52
a = 6.25 – 2.25
a = 4 in
Correct Process
Validation
Chapter 3 — Geometry
194
Identify Errors
or Validate
Worked Solution
4) Find the area of triangle
DEF.
D
e = 8.5
f=5
E
h=4
d = 10.5
F
A = ½bh
A = (½)(10.5)(5)
A = 26.25 square units
5) A right triangle has
∠1 = 26°. What is the
measure of ∠2?
1
2
The triangle is a right triangle,
with one angle = 90°.
The three angles must add to
180°, therefore:
180° – (90° + 26°) =
180° – 116° = 64°
∠2 = 64°
Correct Process
Validation