Download 10-1 Introduction to Analytic Geometry

Advanced Mathematical Concepts
Chapter 10
Lesson 10-1
Example 1
Find the distance between points (-2, 8) and (6, 2).
d=
d=
d=
d=
(x2 - x1)2 + (y2 - y1)2
(6 - (-2))2 + (2 - 8)2
82 + (-6)2
100 or 10
Distance Formula
Let (x1, y1) = (-2, 8) and (x2, y2) = (6, 2).
The distance is 10 units.
Example 2
HIDE AND SEEK Two children are playing hide and seek on the school playground. One of the
children is hiding at the location (-4, 3) on a map grid of the playground. The child who is doing the
seeking is currently at the location (5, -2). Each side of a square on the grid represents 5 feet. How
far apart are the two children?
Use the distance formula to find the distance between (-4, 3) and (5, -2).
d=
d=
d=
d=
(x2 - x1)2 + (y2 - y1)2
(5 - (-4))2 + (-2 - 3)2
92 + (-5)2
106 or about 10.3
Distance Formula
Let (x1, y1) = (-4, 3) and (x2, y2) = (5, -2).
The map distance is 10.3 units. Each unit equals 5 feet. So, the actual distance is about 5(10.3) or 51.5
feet.
Advanced Mathematical Concepts
Chapter 10
Example 3
Determine whether quadrilateral ABCD with vertices A(5, 3), B(4, -2), C(-1, -2), and D(0, 3) is a
parallelogram.
Recall that a quadrilateral is a parallelogram if one
pair of opposite sides are parallel and congruent.
First, graph the figure. DA and CB are one pair of
opposite sides.
To determine if the two sides are parallel, find their slopes.
slope of DA
y2 - y1
m =x -x
2
1
3-3
=
5-0
=0
Slope formula
Let (x1, y1) = (0, 3) and (x2, y2) = (5, 3).
slope of CB
y2 - y1
m =x -x
2
1
-2 - (-2)
= 4 - (-1)
=0
Slope formula
Let (x1, y1) = (-1, -2) and (x2, y2) = (4, -2).
Their slopes are equal. Therefore, DA  CB .
To determine if the two sides are congruent, use the distance formula.
DA = (x2 - x1)2 + (y2 - y1)2
= (5 - 0)2 + (3 - 3)2
= 25 or 5
CB = (x2 - x1)2 + (y2 - y1)2
= [4 - (-1)]2 + [-2 - (-2)]2
= 25 or 5
The measures of the two sides are equal. Therefore, they are congruent.
Since the opposite sides are parallel and congruent, quadrilateral ABCD is a parallelogram.
Advanced Mathematical Concepts
Chapter 10
Example 4
Find the coordinates of the midpoint of the segment that has endpoints at (-4, 8) and (5, 3).
Let (-4, 8) be (x1, y1) and (5, 3) be (x2, y2). Use the Midpoint Formula.
x1 + x2 y1 + y2
 2 , 2 
-4 + 5 8 + 3
= 2 , 2 
1 11
= , 
2 2
1 11
The midpoint of the segment is at  , .
2 2
Example 5
Prove that the diagonals of a rectangle are congruent.
In rectangle ABCD, use the Distance Formula to find AC
and BD.
AC = (x2 - x1)2 + (y2 - y1)2
= (a - 0)2 + (b - 0)2
= a2 + b2
BD = (x2 - x1)2 + (y2 - y1)2
= (0 - a)2 + (b - 0)2
= a2 + b2
Since both AC and BD are equal to a2 + b2, AC  BD . Thus, the diagonals of a rectangle are
congruent.