Math 122!
Coordinate Geometry! Prerequisite: None
Course Description
In this micro-course, we’ll work through practice problems involving coordinates in one dimension (calculating distances on horizontal and vertical lines) and two dimensions (calculating length of diagonal lines, identifying missing coordinates, using coordinates to find area/perimeter of basic geometric figures) and slopes (including parallel and perpendicular lines).!
Math 122
2
Coordinate Geometry Problems
Coordinate Geometry Problems! ! Identify:! If you see a coordinate plane, or the question involves (x, y) coordinates.! ! Set Up:! Start plotting points, which usually yields a line or a geometric figure.! ! Make Sure:! Be careful when plotting your points. The x-axis is horizontal, and the y-axis is vertical.! ! Execute:! Apply the appropriate geometry formula and do the math.!
Math 122
3
Coordinate Geometry Problems
Example:! In the xy-coordinate plane, what is the distance between the points with coordinates (2, 5) and (–3, 3)? (A)
2
(B)
29
(C)
45
(D)
65
(E)
89
Math 122
4
Slope Problems
Slope Problems! ! Identify:! You will be given TWO POINTS on a line in the xy-coordinate plane and you will be asked to ! find the slope.! ! Set Up:! rise y 2 − y1 Use the slope formula:! SLOPE = = run x 2 − x1 ! ! Make Sure:! Don’t mix up the x and y coordinates. And remember that the y-values go on TOP of the fraction.! ! Execute:! Do the math!!
Math 122
5
Slope Problems
Example:! y
4 2 –4 –2
0
2
4
x
–2 –4
What is the slope of the line shown in this graph?
Math 122
/
/
0
0
0
1
1
1
1
2
2
2
2
3
3
3
3
4
4
4
4
5
5
5
5
6
6
6
6
7
7
7
7
8
8
8
8
9
9
9
9
6
Slope-Intercept Problems
Slope-Intercept Problems! ! Identify:! You will be given the EQUATION OF A LINE in the xy-coordinate plane.! ! Set Up:! Use the slope-intercept form to identify the slope (m) and y-intercept (b): y = mx + b! ! Make Sure:! If the problem asks for a PARALLEL line, slopes are equal.! If the problem asks for a PERPENDICULAR line, slopes are negative reciprocals. ! ! Execute:! Solve for the question being asked.!
Math 122
7
Slope-Intercept Problems
Example:!
y
O
x
Which of the following could be the equation of the line that passes through the two points in the xy-plane above? (A) y = 2x + 5 (B) y = – 2x – 5 (C) y = – x (D) y =
1 2
x–5
(E) y = – 1 x + 5 2
Math 122
8
Review Questions The points V, W, Y, and Z lie on a line, though not necessarily in that order. Point Y is the midpoint of VW, and VW = 16. If YZ = 4, what are all the possible distances between the points V and Z along this line? (A) 4 (B) 8 (C) 12 (D) 4 and 8 (E) 4 and 12
Math 122
9
Review Questions The equation of line m in the xy-plane is y = 4x – 2. If line n is the reflection of line m over the x-axis, what is the equation of line n ? (A) (B) (C) (D) (E)
y = 4x + 2 y = –4x + 2 y = –4x – 2 y = 0.25x + 2 y = –0.25x – 2
Math 122
10
Review Questions Y
9
A
7
B
8
C
4
D
8
E
l
Note: Figure not drawn to scale.
In the figure above, YC is perpendicular to l. Which of the following line segments (not shown) has the greatest length? (A) YA (B) YB (C) YC (D) YD (E) YE
Math 122
11
Review Questions The points A, B, C, and D all lie on the same line in such a way that C is the midpoint of AB and D is the midpoint of AC. If AB = 12, what is the length of BD? (A) 3 (B) 4 (C) 6 (D) 8 (E) 9
Math 122
12
Review Questions y 3 L 0
M 3
x
Note: Figure not drawn to scale.
In the figure above, line r (not shown) is perpendicular to segment LM and bisects segment LM. Which of the following points lies on line r? (A) (B) (C) (D) (E)
(0, 1) (1, 2) (2, 1) (2, 2) (2, 4)
Math 122
13
Review Questions y (3,8) (–1,w) O
x
Note: Figure not drawn to scale.
In the figure above, the slope of the line is 1.5. What is the value of w? (A) –3 (B) –2 (C) 0 (D) 1 (E) 2
Math 122
14
Review Questions Rectangle PQRS lies in the xy-coordinate plane so that its sides are not parallel to the axes. What is the product of the slopes of all four sides of rectangle PQRS? (A) –2 (B) –1 (C) 0 (D) 1 (E) 2
Math 122
15
Review Questions Y
Z
–11
5
On the number line above, what is the coordinate of the point between Y and Z that is three times as far from point Z as from point Y ? (A) (B) (C) (D) (E)
–9 –7 –3 –1 1
Math 122
16
Review Questions Which of the following is the slope of a line that is perpendicular to the line 3x + 5y = 10? (A) (B) (C)
5 3 3 5 1 2
3 5 5 (E) 3
(D)
Math 122
17
Review Questions y B (n, 7) A (1, 1) 0
x
Note: Figure not drawn to scale.
In the figure above, the slope of the line through points A and B is (A) (B) (C) (D) (E)
3 . What is the value of n? 2
4 5 6 8 9
Math 122
18
Review Questions y m
(0, r)
(p, 0) 0
x
Which of the following is the slope of a line that is perpendicular to the line 3x + 5y = 10? r p p r
(A) (B) (C) 0 r
(D) p (E)
p r
Math 122
19
Review Questions y 4 3 2 1 0
1 2 3 4
x l
What is the slope of line that is perpendicular to line l shown in the figure above? (A) –3 (B)
3 2
(C) 0 (D)
2 3
(E) 3
Math 122
20
Review Questions In the (x, y) coordinate plane, a line passes through the point 1 (–4, 6) and has a slope of . What is the x-coordinate of a 3 point on the line having a y-coordinate of 3? (A) (B) (C) (D) (E)
–13 –8 3 5 9
Math 122
21
Review Questions y O (0,0)
x (– 5,– 3) l
Q R
Note: Figure not drawn to scale. Line s (not shown) passes through O and intersects line segment QR between Q and R. What is one possible value of the slope of line s ? 3 5
(A)
(B) –1 (C) 0 (D)
1 4
(E) 5
Math 122
22
Review Questions y (0, 4)
(5, 0)
x
What is the area of the triangle in the coordinate plane above? (A) (B) (C) (D) (E)
4.5 9.0 10.0 12.5 20.0
Math 122
23