SECTION 1.1 Points, Regions, Distance, and Midpoints
Chapter 1
An Introduction to Graphs and Lines
Section 1.1:
Points, Regions, Distance and Midpoints
Points in the Coordinate Plane
Regions in the Coordinate Plane
The Distance Formula
The Midpoint Formula
Points in the Coordinate Plane
MATH 1310 College Algebra
1
CHAPTER 1 An Introduction to Graphs and Lines
Solution:
Additional Example 1:
2
University of Houston Department of Mathematics
SECTION 1.1 Points, Regions, Distance, and Midpoints
Solution:
MATH 1310 College Algebra
3
CHAPTER 1 An Introduction to Graphs and Lines
Additional Example 2:
Solution:
4
University of Houston Department of Mathematics
SECTION 1.1 Points, Regions, Distance, and Midpoints
Additional Example 3:
Solution:
Regions in the Coordinate Plane
MATH 1310 College Algebra
5
CHAPTER 1 An Introduction to Graphs and Lines
{( x, y )
− 3 < y ≤ 2}
4
3
2
1
−2
−1
x
1
2
3
−1
−2
−3
−4
−5
The region consists of all points that:
(a) lie on the line y = 2;
(b) lie between the lines y = 2 and y = −3.
The broken line y = −3 indicates that points
on this line do not lie in the region.
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University of Houston Department of Mathematics
SECTION 1.1 Points, Regions, Distance, and Midpoints
{( x, y ) 1 < x < 4}
2
1
−1
1
2
3
4
5
−1
−2
The region consists of all points that lie
between the lines x = 1 and x = 4. The
broken lines at x = 1 and x = 4 indicate that
points on this line do not lie in the region.
Solution:
Additional Example 1:
Solution:
MATH 1310 College Algebra
7
CHAPTER 1 An Introduction to Graphs and Lines
Additional Example 2:
Solution:
3
2
y=1
1
−3
−2
−1
1
2
3
−1
−2
8
University of Houston Department of Mathematics
SECTION 1.1 Points, Regions, Distance, and Midpoints
Additional Example 3:
Solution:
5
4
y=3
3
2
1
−2
−1
1
2
−1
−2
y = −2
−3
−4
−5
Additional Example 4:
Solution:
MATH 1310 College Algebra
9
CHAPTER 1 An Introduction to Graphs and Lines
5
4
y=3
3
2
1
−4
−3
−2
−1
1
2
3
4
−1
−2
x = −1
The Distance Formula
10
University of Houston Department of Mathematics
SECTION 1.1 Points, Regions, Distance, and Midpoints
Example Problem: Find the distance between the points A (3, − 2) and B (−1, 3) .
Solution:
Additional Example 1:
Solution:
Additional Example 2:
Solution:
MATH 1310 College Algebra
11
CHAPTER 1 An Introduction to Graphs and Lines
Additional Example 3:
Solution:
Additional Example 4:
Solution:
12
University of Houston Department of Mathematics
SECTION 1.1 Points, Regions, Distance, and Midpoints
MATH 1310 College Algebra
13
CHAPTER 1 An Introduction to Graphs and Lines
14
University of Houston Department of Mathematics
SECTION 1.1 Points, Regions, Distance, and Midpoints
Additional Example 5:
Solution:
MATH 1310 College Algebra
15
CHAPTER 1 An Introduction to Graphs and Lines
16
University of Houston Department of Mathematics
SECTION 1.1 Points, Regions, Distance, and Midpoints
Additional Example 6:
Solution:
MATH 1310 College Algebra
17
CHAPTER 1 An Introduction to Graphs and Lines
Additional Example 7:
Solution:
18
University of Houston Department of Mathematics
SECTION 1.1 Points, Regions, Distance, and Midpoints
Use the Pythagorean Theorem to determine c.
The Midpoint Formula
MATH 1310 College Algebra
19
CHAPTER 1 An Introduction to Graphs and Lines
Solution:
Additional Example 1:
Solution:
Additional Example 2:
Solution:
20
University of Houston Department of Mathematics
SECTION 1.1 Points, Regions, Distance, and Midpoints
Additional Example 3:
Solution:
Additional Example 4:
Solution:
MATH 1310 College Algebra
21
CHAPTER 1 An Introduction to Graphs and Lines
Additional Example 5:
22
University of Houston Department of Mathematics
SECTION 1.1 Points, Regions, Distance, and Midpoints
Solution:
MATH 1310 College Algebra
23
CHAPTER 1 An Introduction to Graphs and Lines
24
University of Houston Department of Mathematics
Exercise Set 1.1: Points, Regions, Distance and Midpoints
17. Given the following points:
(-3, 4), (0, 4), (1, 4), (3, 4)
Plot the following points on a coordinate plane.
1.
A(3, 4)
2.
B(-2, 5)
3.
C(-3, -1)
4.
D(-5, 0)
5.
E(-4, -6)
6.
F(0, -2)
(a)
(b)
(c)
(d)
Graph the following regions in a coordinate plane.
18.
Write the coordinates of each of the points shown in
the figure below. Then identify the quadrant (or axis)
in which the point is located.
7.
G
6
8.
H
4
9.
I
21.
I
22.
H
G
2
10. J
J
23.
0
-6
12. L
-4
-2
-2
0
19.
20.
y
11. K
Plot the above points on a coordinate plane.
What do the above points have in common?
Draw a line through the above points.
What is the equation of the line drawn in
part (c)?
2
4
6
24.
25.
K -4
L
-6
26.
{ ( x, y )
{ ( x, y )
{ ( x, y )
{ ( x, y )
{ ( x, y )
{ ( x, y )
{ ( x, y )
{ ( x, y )
{ ( x, y )
x=2}
y = −5 }
y >2}
x ≤3}
−1 < x ≤ 4 }
−3 ≤ y < 5 }
x ≤ 1 and y > −3 }
x > 4 and y < −2 }
x ≥ 2 and y ≥ 1 }
Answer the following.
Area of a rectangle = base x height
Area of a parallelogram = base x height
Area of a triangle = ½ (base x height)
13. Draw the rectangle with vertices A(3, 5), B(-2,5),
C(-2, -4), and D(3, -4). Then find the area of
rectangle ABCD.
Use the Pythagorean Theorem to find the missing side
of each of the following triangles.
Pythagorean Theorem: In a right triangle, if a and b
are the measures of the legs, and c is the measure of
the hypotenuse, then a2 + b2 = c2.
c
a
14. Draw the parallelogram with vertices A(-2, -3),
B(4, -3), C(6, 1), and D(0, 1). Then find the area
of parallelogram ABCD.
15. Draw the triangle with vertices E(-3, 2), F(4, 2),
and G(1, 5). Then find the area of triangle EFG.
b
27.
Answer the following.
16. Given the following points:
(3, 5), (3, 1), (3, 0), (3, -2)
(a)
(b)
(c)
(d)
Plot the above points on a coordinate plane.
What do the above points have in common?
Draw a line through the above points.
What is the equation of the line drawn in
part (c)?
MATH 1310 College Algebra
c
5
12
7
28.
a
5
25
Exercise Set 1.1: Points, Regions, Distance and Midpoints
Answer the following.
29. Given the following points:
A(1, 2) and B(4, 7)
(a) Plot the above points on a coordinate plane.
(b) Draw segment AB. This will be the
hypotenuse of triangle ABC.
(c) Find a point C such that triangle ABC is a
right triangle. Draw triangle ABC.
(d) Use the Pythagorean theorem to find the
distance between A and B (the length of the
hypotenuse of the triangle).
30. Given the following points:
A(-3, 1) and B(1, -5)
(a) Plot the above points on a coordinate plane.
(b) Draw segment AB. This will be the
hypotenuse of triangle ABC.
(c) Find a point C such that triangle ABC is a
right triangle. Draw triangle ABC.
(d) Use the Pythagorean theorem to find the
distance between A and B (the length of the
hypotenuse of the triangle).
Use the distance formula to find the distance between
the two given points. (You may also use the method from
the previous two problems to verify your answer.)
31. (3, 6) and (5, 9)
32. (4, 7) and (2, 3)
33. (-5, 0) and (-2, 6)
34. (9, -4) and (2, -3)
35. (4, 0) and (0, -7)
36. (-4, -7) and (-10, -2)
Answer the following.
43. If M(3, -5) is the midpoint of the line segment
joining points A and B, and A has coordinates
(-1, -2), find the coordinates of B.
44. If M(-2, 4) is the midpoint of the line segment
joining points A and B, and A has coordinates
(-1, -2), find the coordinates of B.
45. Determine which of the following points is
closer to the origin: A(5, -6) or B(-3, 7)?
46. Determine which of the following points is
closer to the point (4, -1): A(-2, 3) or B(6, 6)?
47. Determine whether or not the triangle formed by
the following vertices is a right triangle:
A(3, 2), B(8, 5), and C(6, 10)
48. Determine whether or not the triangle formed by
the following vertices is a right triangle:
A(4, 3), B(5,0), and C(-1, -2)
49. Determine if the points A(-5,2), B(-1, 8), and
C(1, 11) are collinear.
50. Determine if the points A(-1,3), B(2, 4), and
C(6, 5) are collinear.
51. Given the points A(-3,2), B(2, 5), and C(9, 4),
determine the point D so that ABCD is a
parallelogram.
52. A circle has a diameter with endpoints A(-5, -9)
and B(3, 5).
(a) Find the coordinates of the center of the
circle.
(b) Find the length of the radius of the circle.
Find the midpoint of the line segment joining points A
and B.
37. A(7, 6) and B(3, 8)
38. A(5, 9) and B(1, 3)
39. A(-7, 0) and B(-4, 8)
40. A(7, -5) and B(4, -3)
41. A(3, 0) and B(0, -9)
42. A(-6, -7) and B(-10, -6)
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University of Houston Department of Mathematics