Geometry
Coordinate Geometry (1)
Name______________________
The Basic Tools of Coordinate Geometry
In the 1600’s Rene Descartes developed the coordinate plane (which is why we refer to the Cartesian plane), allowing mathematicians to think about geometric relationships within an algebraic context. In this unit we will revisit some of the theorems we have studied in a new context, with figures placed on the Cartesian plane. The following algebraic relationships will be especially useful.
The Distance Formula
What is d, the distance from A to B? d
B ( x
2
, y
The Slope Formula
What is m, the left to right steepness of the line that contains A and B?
Slopes of Parallel Lines
Why do parallel lines have equal slopes?
b
A ( x
1
, y
1
)
Slopes of Perpendicular Lines
What is the relationship between slopes of perpendicular lines?
a
a b
The Midpoint Formula
What are the coordinates of point M, a point that bisects a segment AB?
A ( x
1
, y
1
)
Linear Equation
What is the equation of a line with slope m that passes through point (x
1
, y
1
)?
M
B ( x
2
, y
2
)
2
)

Geometry
Coordinate Geometry (1)
Exercises
1. P has coordinates (−2, 2) and Q has (8, 2). a. Find the slope of PQ b. Find the coordinates of the midpoint of PQ. c. Find the length of PQ
2. A rectangle has vertices at (4, 0), (0,0), and
(0,3). a. Find the coordinates of the fourth vertex b. Find the length of each diagonal. c. Find the coordinates of the intersection
point of the diagonals.
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Geometry
Coordinate Geometry (1)
3. Point A has coordinates (3, 2)
Point B has (−9, −3), and Point C has (8, −10). a. Find the size of angle ABC. b. Find the lengths of AB and AC.
AB
2 
AC
2 
BC
2 c.
Does ?
What does this say about 3-gon ABC?
4. Points A, B, C, D have coordinates (1, 5), (6, 8),
(8, 6) and (3, 3).
Show that AB = DC and AD = BC.
What type of quadrilateral is ABCD?
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Geometry
Coordinate Geometry (1)
5. Given segment AB with A: (4, 0) and B: (-4, 8) a. Find the coordinates of the midpoint of AB. b. Find the slope of AB. c. Find the slope of a line perpendicular to AB. d. Find the equation of the line perpendicular
to AB through the midpoint. d.
Given your equation in part d, if x = 1, find
the value of y.
6. Determine whether the three points with coordinates below are collinear: a. A: (0, -1) B: (1, 1) C: (2, 3) b. D: (0, 2) E: (2, 5) F: (3, 7) c. G: (-1, 4) H: (2, 1) I: (-2, 5)
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Geometry
Coordinate Geometry (1)
7. A line segment has one end at (5, 4) and the
midpoint at (12, 10). Where is the other end?
8. The following are the coordinates of the vertices of  EFG: E: (−1, −3), F: (2, −3),
G: (2, 1) a. Find the lengths of EF and FG. b. Find the distance from E to G. c. Find the area of  EFG.
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Geometry
Coordinate Geometry (1)
9. Find equations for the following lines: a. Passing through (2,3) and (5, −6). b. Parallel to the line y = 4x − 3 but passing
through the point (2,6). c. With slope 15 passing through the origin.
10. To “trisect” anything is to break it into three equal pieces. a. Given line segment AB where A is (2, 5) and B is (20, 26), find the two points that trisect it. Drawing the points may help. b. Find a formula for trisecting any line segment AB where A is (x
1
, y
1
) and B is (x
2
, y
2
) .
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