Transcript Interactive Chalkboard - Tuslaw Local School District
Transformations on the Coordinate Plane
Area: Parallelograms, Triangles, and Trapezoids
Circumference and Area: Circles
Example 1 Find Measures of Angles
Example 2 Find a Missing Angle Measure
Example 3 Find Measures of Angles
Example 4 Apply Angle Relationships
In the figure,
m
find
m
2
and
||
n m
8
.
and
t
is a transversal. If
Since are alternate exterior angles, they are congruent. So, .
Answer:
Since are corresponding angles, they are congruent. So, .
Answer:
In the figure,
m
find
m
5
and
||
n m
1
.
and
t
is a transversal. If Answer:
Multiple-Choice Test Item If and
D
and what is
m
E
?
E
are complementary, A
53
°
C
127
°
B
37
°
D
7
°
Read the Test Item
Since are complementary, .
Solve the Test Item Answer:
The answer is B
.
Complementary angles Replace with
53
°
.
Subtract
53
from each side.
Multiple-Choice Test Item If and
G
and what is
m
h
?
A
76
°
B
104
°
H
C
83
°
are supplementary, D
14
°
Answer:
A
Angles
PQR
and
STU
and are supplementary. If , find the measure of each angle.
Step 1
Find the value of
x
.
Supplementary angles Substitution Combine like terms.
Add
80
to each side.
Divide each side by
2
.
Step 2
Replace
x
each angle
.
with
130
to find the measure of
Answer:
Angles
ABC
and
DEF
and are complementary. If , find the measure of each angle.
Answer:
Transportation A road crosses railroad tracks at an angle as shown. If find
m
6
and
m
5
.
Since are corresponding angles, they are congruent.
Answer:
Since are supplementary angles, the sum of their measures is
180
°
.
180 – 131 = 49
Answer:
Transportation Main Street crosses Broadway Boulevard and Maple Avenue at an angle as shown. If
m
1 = 48
°
, find
m
3
and
m
4
.
Answer:
Example 1 Name Corresponding Parts
Example 2 Use Congruence Statements
Example 3 Find Missing Measures
Name the corresponding parts in the congruent triangles shown. Then complete the congruence statement.
Answer:
Corresponding Angles Corresponding Sides One congruence statement is
.
HGI
?
Name the corresponding parts in the congruent triangles shown. Then complete the congruence statement.
ABC
?
Answer:
If , complete each congruence statement.
Explore Plan
You know the congruence statement. You need to find the corresponding parts.
Use the order of the vertices in to identify the corresponding parts.
Solve
Answer:
M
corresponds to
Q
, and
N
corresponds to
P
so
N
corresponds to
P
, and
O
corresponds to
R
so
M
corresponds to
Q
, and
O
corresponds to
R
so
Examine
Draw the triangles, using arcs and slash marks to show the congruent angles and sides.
If statement.
, complete each congruence Answer:
Construction In the figure, of
F
? A brace is used to support a tabletop. . What is the measure
F
and
C
are corresponding angles. So, they are congruent. Since .
Answer:
What is the length of ?
corresponds to Since ,
Answer:
. So, .
and are congruent.
Art In the figure, . a.
What is the measure of
B
?
Answer:
44
°
b.
What is the length of
Answer:
22
in.
?
Example 1 Translation in a Coordinate Plane
Example 2 Reflection in a Coordinate Plane
Example 3 Rotations in a Coordinate Plane
The vertices of
ABC
are
A
(–3
,
7)
,
B
(–1
,
0)
, and
C
(5
,
5)
. Graph the triangle and the image of
ABC
after a translation
4
units right and
5
units down.
This translation can be written as the ordered pair
(4
,
–5)
.
To find the coordinates of the translated image, add
4
each
x
-coordinate and add
–5
to each
y
-coordinate.
to
vertex
A
(–3
,
7)
B
(–1
,
0)
C
(5
,
5)
4 right, 5 down
(4
,
–5) (4
,
–5) (4
,
–5)
translation
A
(1
,
2)
B
(3
,
–5)
C
(9
,
0)
The coordinates of the vertices of
B
(3
,
–5)
, and
C
(9
,
0)
. Graph
A
B
C
ABC
are
and
A
(1
,
2)
,
A
B
C
.
Answer:
The vertices of
DEF
are
D
(–1
,
5)
,
E
(–3
,
1)
, and
F
(4
, –
4)
. Graph the triangle and the image of
D
E
F
after a translation
3
units left and
2
units up.
Answer:
The vertices of a figure are
M
(–8
,
6)
,
N
(5
,
9)
,
O
(2
,
1)
, and
P
(–10
,
3)
. Graph the figure and the image of the figure after a reflection over the
y
-axis.
To find the coordinates of the vertices of the image after a reflection over the
y
-axis, multiply the
x
-coordinate by
–1
and use the same
y
-coordinate.
vertex
M
(–8
,
6)
N
(5
,
9)
O
(2
,
1)
P
(–10
,
3)
reflection
M
(8
,
6)
N
(–5
,
9)
O
(–2
,
1)
P
(10
,
3)
The coordinates of the vertices of the reflected figure are
M
(8
,
6)
,
N
(–5
,
9)
,
O
(–2
,
1)
and
P
(10
,
3)
. Graph the figure and its image.
Answer:
4 –8 –4 –4 4 8
The vertices of a figure are
Q
(–2
,
4)
,
R
(–3
,
1)
,
S
(3
,
–2)
, and
T
(4
,
3)
. Graph the figure and the image of the figure after a reflection over the
y
-axis.
Answer:
A figure has vertices
A
(–4
,
5)
,
B
(–2
,
4)
,
C
(–1
,
2)
,
D
(–3
,
1)
, and
E
(–5
,
3)
. Graph the figure and the image of the figure after a rotation of
180
°
.
To rotate the figure, multiply both coordinates of each point by
–1
.
A B C D E
(–4 (–2 (–1 (–3 (–5
, , , , ,
5) 4) 2) 1) 3)
A D
(4
B
(2
,
C
(1
,
E
(3 (5
, , ,
–5) –4) –2) –1) –3)
The coordinates of the vertices of the rotated figure are
A
(4
,
–5)
,
B
(2
,
–4)
,
C
(1
,
–2)
,
D
(3
,
–1)
, and
E
(5
,
–3)
. Graph the figure and its image.
Answer:
A figure has vertices
A
(2
,
–1)
,
B
(3
,
4)
,
C
(–3
,
4)
,
D
(–5
,
–1)
, and
E
(1
,
–4)
. Graph the figure and the image of the figure after a rotation of
180
°
.
Answer:
Example 2 Classify Quadrilaterals
Find the value of
x
. Then find each missing angle measure.
Words Variable
The sum of the measures of the angles is
360
° .
Let
m
Q
,
m
R
,
m
S
, and
m
T
the measures of the angles.
represent
Equation Answer:
The value of
x
and is
35
. So, .
Angles of a quadrilateral Substitution Combine like terms.
Subtract
185
from each side.
Simplify.
Divide each side by
5
.
Find the value of
x
. Then find each missing angle measure.
Answer:
Classify the quadrilateral using the name that best describes it.
The quadrilateral has one pair of opposite sides parallel.
Answer:
It is a trapezoid.
Classify the quadrilateral using the name that best describes it.
The quadrilateral has both pairs of opposite sides parallel and congruent.
Answer:
It is a parallelogram.
Classify the quadrilateral using the name that best describes it.
The quadrilateral has four congruent sides and four right angles.
Answer:
It is a square.
Classify each quadrilateral using the name that best describes it.
a.
b.
Answer:
rectangle
c.
Answer:
parallelogram
Answer:
trapezoid
Example 1 Find Areas of Parallelograms
Example 2 Find Areas of Triangles
Example 3 Find Area of a Trapezoid
Example 4 Use Area to Solve a Problem
Find the area of the parallelogram.
The base is
3
The height is
3
meters. meters.
Area of a parallelogram Replace
b h
with
3
.
with
3
and Multiply.
Answer:
The area is
9
square meters.
Find the area of the parallelogram.
The base is
4.3
The height is
6.2
inches. inches.
Area of a parallelogram Replace
b
and
h
with with
6.2
.
4.3
Multiply.
Answer:
The area is
26.66
square inches.
Find the area of each parallelogram.
a.
b.
Answer:
12 cm
2
Answer:
1.95 ft
2
Find the area of the triangle.
The base is
3
The height is
4
meters. meters.
Area of a triangle Replace
b
and
h
with with
4
.
3
Multiply.
Answer:
The area of the triangle is
6
square meters.
Multiply.
Find the area of the triangle.
The base is
3.9
The height is
6.4
feet. feet.
Area of a triangle Replace and
h b
with
3.9
with
6.4
.
Multiply.
Multiply.
Answer:
The area of the triangle is
12.48
square feet.
Find the area of each triangle.
a.
b.
Answer: Answer:
Find the area of the trapezoid.
The height is
6
meters. The bases are meters and meters. Area of a trapezoid Replace
h
with
6
and
a
with and
b
with .
Divide out the common factors.
Simplify.
Answer:
The area of the trapezoid is square meters.
Find the area of the trapezoid.
Answer:
Painting A wall that needs to be painted is
16
feet wide and
9
feet tall. There is a doorway that is
3
feet by
8
feet and a window that is
6
feet by to be painted?
feet. What is the area
To find the area to be painted, subtract the areas of the door and window from the area of the entire wall.
Area of the wall Area of the door Area of the window Answer:
The area to be painted is
144 – 24 – 33
or
87
square feet.
Gardening A garden needs to be covered with fresh soil. The garden is 12 feet wide and 15 feet long. A rectangular concrete path runs through the middle of the garden and is 3 feet wide and 15 feet long. Find the area of the garden which needs to be covered with fresh soil.
Answer:
Example 2 Measures of Interior Angles
Example 3 Find Angle Measure of a Regular Polygon
Classify the polygon.
This polygon has 5 sides.
Answer:
It is a pentagon.
Classify the polygon.
This polygon has 7 sides.
Answer:
It is a heptagon.
Classify each polygon.
a.
b.
Answer:
hexagon
Answer:
heptagon
Find the sum of the measures of the interior angles of a quadrilateral.
A quadrilateral has 4 sides. Therefore, . Replace
n
with
4.
Simplify.
Answer:
The sum of the measures of the interior angles of a quadrilateral is
360
° .
Find the sum of the measures of the interior angles of a pentagon.
Answer:
540
°
Traffic Signs A stop sign is a regular octagon. What is the measure of one interior angle in a stop sign?
Step 1
Find the sum of the measures of the angles. An octagon has
8
sides. Therefore, .
Replace
n
with
8.
Simplify.
The sum of the measures of the interior angles is
1080
° .
Step 2
Divide the sum by
8
of one angle.
to find the measure
Answer:
So, the measure of one interior angle in a stop sign is
135
° .
Picnic Table A picnic table in the park is a regular hexagon. What is the measure of one interior angle in the picnic table?
Answer:
120
°
Example 1 Find the Circumference of a Circle
Example 2 Use Circumference to Solve a Problem
Example 3 Find Areas of Circles
Find the circumference of the circle to the nearest tenth.
Circumference of a circle Replace
d
with
12
.
Simplify. This is the
exact
circumference.
To estimate the circumference, use a calculator.
12
2nd [ ] ENTER 37.69911184
Answer:
The circumference is about
37.7
meters.
Find the circumference of the circle to the nearest tenth.
Circumference of a circle Replace
r
with
7.1
.
Simplify. Use a calculator.
Answer:
The circumference is about
44.6
meters.
Find the circumference of each circle to the nearest tenth.
a.
b.
Answer:
12.6 ft
Answer:
10.1 cm
Landscaping A landscaper has a tree whose roots form a ball-shaped bulb with a circumference of about 110 inches. How wide will the landscaper have to dig the hole in order to plant the tree?
Explore
You know the circumference of the roots of the tree. You need to know the diameter of the hole to be dug.
Plan Solve
Use the formula for the circumference of a circle to find the diameter.
Circumference of a circle Replace
C
with
110.
Divide each side by
.
Simplify. Use a calculator.
Answer:
The diameter of the hole should be at least
35
inches.
Examine
Check the reasonableness of the solution by replacing
d
with
35
in . Circumference of a circle Replace
d
with
35.
Simplify. Use a calculator.
The solution is reasonable.
Swimming Pool A circular swimming pool has a circumference of 24 feet. Matt must swim across the diameter of the pool. How far will Matt swim?
Answer:
about
7.6
ft
Find the area of the circle. Round to the nearest tenth.
Area of a circle Replace
r
with
11
.
Evaluate . Use a calculator.
Answer:
The area is about
380.1
square feet.
Find the area of the circle. Round to the nearest tenth.
Area of a circle Replace
r
with
4.15
.
Evaluate . Use a calculator.
Answer:
The area is about
54.1
square centimeters.
Find the area of each circle. Round to the nearest tenth.
a.
b.
Answer: Answer:
Example 1 Find Area of Irregular Figures
Example 2 Use Area of Irregular Figures
Find the area of the figure to the nearest tenth.
Explore
You know the dimensions of the figure. You need to find its area.
Plan
Solve a simpler problem.
First, separate the figure into a triangle, square, and a quarter-circle. Then find the sum of the areas of the figure.
Find the area of the figure to the nearest tenth.
Solve
Area of Triangle Area of a triangle Replace
b
and
h
with with
4.
2
Simplify.
Find the area of the figure to the nearest tenth.
Solve
Area of Square Area of a square Replace
b
and
h
with
2.
Simplify.
Find the area of the figure to the nearest tenth.
Solve
Area of Quarter-circle Area of a quarter-circle Replace
r
with
2.
Simplify.
Answer:
The area of the figure is or about
11.1
square inches.
Find the area of the figure to the nearest tenth.
Answer:
Carpeting Carpeting costs $2 per square foot. How much will it cost to carpet the area shown?
Step 1
Find the area to be carpeted.
Area of Rectangle Area of a rectangle Replace
b
and
h
with with
10.
14
Simplify.
Carpeting Carpeting costs $2 per square foot. How much will it cost to carpet the area shown?
Area of Square Area of a square Replace
b
and
h
with
3.
Simplify.
Carpeting Carpeting costs $2 per square foot. How much will it cost to carpet the area shown?
Area of Triangle Area of a triangle Replace
b
and
h
with with
12.
14
Simplify.
The area to be carpeted is or
233
square feet.
Carpeting Carpeting costs $2 per square foot. How much will it cost to carpet the area shown?
Step 2
Find the cost of the carpeting.
Answer:
So, it will cost $466 to carpet the area shown.
Painting One gallon of paint is advertised to cover 100 square feet of wall surface. About how many gallons will be needed to paint the wall shown below?
Answer:
about 4 gallons