Objective 3
The student will demonstrate an understanding of geometry and spatial reasoning.
For this objective you should be able to
●
identify essential attributes of two-dimensional and
three-dimensional geometric figures;
●
describe the results of transformations; and
●
locate and name points on a coordinate grid.
What Are Parallel Lines?
Parallel lines are lines that are the same distance apart at all points.
Parallel lines never intersect. Several examples of parallel lines are
shown below.
B
A
R
J
L
K
M
S
D
C
T
V
You can also look at a figure and identify line segments that are
parallel. Line segments in figures are parallel if they are the same
distance apart at all points.
Which two line segments in the figure below appear to be parallel?
W
X
Z
Y
Line segment WX and line segment ZY appear to be the same
distance apart at all points. Line segments WX and ZY appear to be
parallel.
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Line AB can also be
named line BA.
Objective 3
What Are Perpendicular Lines?
Perpendicular lines are lines that intersect to form right angles.
Angle GXF is a right angle. Line EF is perpendicular to line GH.
G
A small
is placed at
the vertex of an angle to
show that it is a right
angle.
E
F
X
H
T
X
A
X
S
B
C
You can also look at a figure and identify line segments that are
perpendicular.
Which line segments in the rectangle below are perpendicular?
A
B
D
C
The rectangle shows a right angle at each of the vertices: A, B, C, and
D. The two line segments that meet at each vertex form a right angle
and are perpendicular to each other.
●
Line segments AB and BC are perpendicular.
●
Line segments BC and CD are perpendicular.
●
Line segments CD and AD are perpendicular.
●
Line segments AD and AB are perpendicular.
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Objective 3
Try It
Look at square KLMN below.
K
L
N
M
Which line segments are parallel?
Opposite sides of a square are parallel. Parallel lines never
_______________ or _______________.
Line segment _______ and line segment _______ are parallel.
Line segment _______ and line segment _______ are parallel.
Which line segments are perpendicular?
There is a ___________ angle at each vertex of a square.
Perpendicular lines intersect at right angles.
Line segment _______ and line segment _______ are perpendicular.
Line segment _______ and line segment _______ are perpendicular.
Line segment _______ and line segment _______ are perpendicular.
Line segment _______ and line segment _______ are perpendicular.
Parallel lines never intersect or cross. Line segment KL and line segment
NM are parallel. Line segment KN and line segment LM are parallel. There is
a right angle at each vertex of a square. Line segment KL and line segment
KN are perpendicular. Line segment KL and line segment LM are
perpendicular. Line segment NM and line segment LM are perpendicular.
Line segment NM and line segment KN are perpendicular.
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Objective 3
How Can You Describe a Two-Dimensional Figure?
You can describe a two-dimensional figure (or plane figure) by
counting the number of sides, vertices, and angles the figure has.
Do you see
that . . .
Polygons are closed two-dimensional figures with straight sides. A
circle is a closed two-dimensional figure, but it is not a polygon
because it has no sides. You should be able to recognize and describe
the following polygons.
Polygons
Figure
Description
Congruent
Sides
3 sides
3 vertices
● 3 angles
●
Triangle
●
All sides equal
in length
A rectangle is a
quadrilateral with
opposite sides congruent
and 4 right angles. A
square is a rectangle with
all sides congruent.
4 sides
4 vertices
● 4 angles
●
Quadrilateral
●
All sides equal
in length
5 sides
5 vertices
● 5 angles
●
Pentagon
●
All sides equal
in length
6 sides
6 vertices
● 6 angles
●
Hexagon
●
All sides equal
in length
8 sides
8 vertices
● 8 angles
●
Octagon
●
All sides equal
in length
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Noncongruent
Sides
Objective 3
Try It
Look at the figures below. Count the number of sides, vertices, and
angles. Then name the figure.
A
●
B
C
D
Figure A has _______ sides, _______ vertices, and _______ angles.
Opposite sides appear to be ______________.
Figure A is a ______________.
●
Figure B has _______ sides, _______ vertices, and _______ angles.
Figure B is a ______________.
●
Figure C has _______ sides, _______ vertices, and _______ angles.
All sides appear to be ______________.
Figure C is a ______________.
●
Figure D has _______ sides, _______ vertices, and _______ angles.
Figure D is a ______________.
Figure A has 4 sides, 4 vertices, and 4 angles. Opposite sides appear to be
congruent. Figure A is a rectangle.
Figure B has 5 sides, 5 vertices, and 5 angles. Figure B is a pentagon.
Figure C has 4 sides, 4 vertices, and 4 angles. All sides appear to be
congruent. Figure C is a square.
Figure D has 6 sides, 6 vertices, and 6 angles. Figure D is a hexagon.
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Objective 3
How Can You Describe a Three-Dimensional Figure?
A three-dimensional
figure has length, width,
and height.
You can describe a three-dimensional figure (or solid figure) by
counting the number of vertices, edges, and faces the figure has.
Edge
Face
Vertex
●
A face is a flat surface in the shape of a two-dimensional
figure.
●
An edge is a line segment where two faces meet.
●
A vertex is a point where three or more edges meet. The
plural of vertex is vertices.
Here are some three-dimensional figures you should be able to
recognize and describe.
Three-Dimensional Figures
Figure
Example
Description
Triangular
prism
●
●
●
●
2
3
9
6
Rectangular
prism
●
●
●
6 rectangular faces
12 edges
8 vertices
Cube
●
●
●
6 square faces
12 edges
8 vertices
●
●
●
●
1
4
8
5
●
●
●
4 triangular faces
6 edges
4 vertices
Square
pyramid
Triangular
pyramid
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triangular faces
rectangular faces
edges
vertices
square face
triangular faces
edges
vertices
Objective 3
Try It
Ernest gave his father a candy bar in a box shaped like the one
below.
e
te
la ar
r
o
a
c
c
b
y b
ho d
ch
dy
n
n
a
ca
c
The box has _______ edges.
The box has _______ vertices.
The box has _______ rectangular faces.
The box has _______ triangular faces.
How many more edges than vertices does this box have?
_______ edges 2 _______ vertices 5 _______
This box has _______ more edges than vertices.
How many faces does this box have?
_______ rectangular faces 1 _______ triangular faces 5 _______
This box has _______ faces.
The box has 9 edges.
The box has 6 vertices.
The box has 3 rectangular faces.
The box has 2 triangular faces.
9 edges 2 6 vertices 5 3. This box has 3 more edges than vertices.
3 rectangular faces 1 2 triangular faces 5 5. This box has 5 faces.
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Objective 3
Some three-dimensional figures have a curved surface. Some of these
figures also have one or two flat circular surfaces called bases.
Curved
surface
Bases
Here are some three-dimensional figures with curved surfaces that you
should be able to recognize and describe.
Three-Dimensional Figures
Figure
Example
Description
Cylinder
●
●
1 curved surface
2 circular bases
Cone
●
●
1 curved surface
1 circular base
Sphere
●
1 curved surface
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Objective 3
What Are Transformations?
Transformations are ways of moving a figure in a plane. Three kinds of
transformations are translations, reflections, and rotations. The result of
a translation, reflection, or rotation is a figure that is congruent to the
original figure.
●
A translation is a sliding movement. A figure can be translated
up, down, left, right, or diagonally by sliding it.
●
A reflection is a mirror image across a line.
●
A rotation is a turning movement around a point. In a rotation,
the figure moves in a circular path.
Rotates
around
this point
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Do you see
that . . .
Objective 3
What transformation is represented in the diagram?
The figure has been moved down. This is a translation.
Try It
Name the transformation in each diagram.
A
B
C
The figures in diagram A show a mirror image. This transformation
is a _________________.
The figures in diagram B show a sliding movement diagonally. This
transformation is a _________________.
The figures in diagram C show a turning movement around a point.
This transformation is a _________________.
The figures in diagram A show a mirror image. This transformation is a reflection.
The figures in diagram B show a sliding movement diagonally. This
transformation is a translation.
The figures in diagram C show a turning movement around a point. This
transformation is a rotation.
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Objective 3
What Is a Coordinate Plane?
A coordinate plane is a grid used to locate points. A point is located by
using an ordered pair of numbers. The two numbers that form the
ordered pair are called the point’s coordinates.
●
Every coordinate plane has a special point called the origin.
The coordinates of the origin are (0, 0).
●
The horizontal line is called the x-axis. The first number of
an ordered pair tells how many units the point is to the right of
the origin. The first number goes right.
●
The vertical line is called the y-axis. The second number of
an ordered pair tells how many units the point is above the
origin. The second number goes up.
Look at the coordinate grid below. The coordinates of point P are
(2, 5).
y
6
P (2, 5)
5
4
3
2
1
Origin
(0, 0)
0
1
2
3
4
5
6
x
The first number of point P is 2. It is 2 units to the right of the
origin.
The second number of point P is 5. It is 5 units above the origin.
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Objective 3
Look at the coordinate grid.
y
6
5
4
C
B
3
A
2
1
0
1
2
3
4
5
6
x
What are the coordinates of points A, B, and C?
●
Starting from the origin, point A is located 4 units to the right
and 2 units up. The ordered pair (4, 2) shows the location of
point A.
●
Starting from the origin, point B is located 2 units to the right
and 4 units up. The ordered pair (2, 4) shows the location of
point B.
●
Starting from the origin, point C is located 1 unit to the right
and 4 units up. The ordered pair (1, 4) shows the location of
point C.
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Objective 3
Try It
Look at the coordinate grid.
y
6
5
R
4
3
2
T
1
S
0
1
2
3
4
5
6
x
What happens when
a point is on the
x-axis or y-axis?
What are the coordinates of points R, S, and T?
Point R is _______ units to the right and _______ units up. The
ordered pair ( _______ , _______ ) shows the location of point R.
Point S is _______ units to the right and _______ units up. The
ordered pair ( _______ , _______ ) shows the location of point S.
Point T is _______ units to the right and _______ units up. The
ordered pair ( _______ , _______ ) shows the location of point T.
Point R is 6 units to the right and 4 units up. The ordered pair (6, 4) shows
the location of point R.
Point S is 3 units to the right and 0 units up. The ordered pair (3, 0) shows
the location of point S.
Point T is 0 units to the right and 2 units up. The ordered pair (0, 2) shows
the location of point T.
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Objective 3
Try It
The grid below shows a map of a neighborhood.
y
9
8
7
Library
6
5
4
BOOKSTORE
3
Store
2
1
0
School
1
2
3
4
5
6
7
8
9
x
The school is _______ units to the right of the origin and _______
unit above the origin. The school is located at ( _______ , _______ ).
The store is _______ units to the right of the origin and _______ units
above the origin. The store is located at (_______ , _______ ).
The library is _______ units to the right of the origin and _______
units above the origin. The library is located at (_______ , _______ ).
The school is 4 units to the right of the origin and 1 unit above the origin.
The school is located at (4, 1).
The store is 6 units to the right of the origin and 3 units above the origin.
The store is located at (6, 3).
The library is 5 units to the right of the origin and 8 units above the origin.
The library is located at (5, 8).
Now practice what you’ve learned.
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