Warm up
 Solve each equation:
 .
x  2x  5  4
 .
3x  4  5
Lesson 10-1 Introduction to
Analytical Geometry
Objective: To find the distance and midpoint
between two points on a coordinate plane
To prove geometric relationships among points
and lines using analytical methods
Analytical Geometry
 The study of coordinate geometry from an
algebraic perspective.
Distance on a Number Line
 Distance between 2 points a and b on a
number line = a  b or b  a
Distance Formula
 If d is the distance between two points
with coordinates (x1, y1) and (x2, y2) then
d
x2  x1    y2  y1 
2
2
Distance Formula
d
x2  x1 2   y2  y1 2
Where d stands for distance
x1 & y1 are one endpoint of a segment
x2 & y2 are the second endpoint of a
segment
(x2 , y2)
(x1 , y1)
d
Example
 Find the distance between points at (4, -2) and
(8, 3).
Midpoint Formula for a Coordinate
Plane
 On a coordinate plane, the coordinates
of the midpoint of a segment whose
endpoints have coordinates (x1,y1) and
(x2,y2) are x1 + x2, y1 + y2
2
2
Midpoint
(x1, y1)
x1 + x2, y1 + y2
2
2
(x2, y2)
Practice
 Find the coordinates of the midpoint of the
segment that has endpoints at (2, 5) and (-4, -7)
Practice
 Determine whether quadrilateral PQRS with
vertices P(-4, 2), Q(-3, -2), R(3, -3) and S(1, 5) is a
parallelogram.
Practice
 Prove that the diagonals of a square are
perpendicular bisectors of each other.
D(0, a)
C(a, a)
A(0, 0)
B(a, 0)