1.7 Midpoint and Distance in the Coordinate Plane
SOL: G3aObjectives: TSW …• To find the midpoint of a segment.• To find the distance between two points in the
coordinate plane.
Midpoint
the point that divides the segment into two congruent () segments.
XA B
Midpoint
If X is the midpoint of AB, then AX XB and AX = XB
How do we find the Midpoint
If we have a Number Line, then we use the two endpoints, add them together and divided by two.
MP Q
a b
Midpoint = a + b 2
Example 1: Finding the Midpoint
The coordinates on a number line of J and K are–12 and 16, respectively. Find the coordinate ofthe midpoint of JK.
J K
-12 16
Midpoint = -12 + 16 2
42
= = 2
Example 2: Finding the Midpoint
MN has the endpoints at -9 and 4. What is the coordinate of its midpoint?
M N
-9 4
Midpoint = -9 + 4 2
-5 2
= = -2 ½
Midpoint Formula: If we are working in the Coordinate Plane
When using the midpoint formula you can use the points in any order, remember addition is commutative.
8 + -14 2
Example 3:
Find the coordinates of the midpoint of GH for
G(8, -6) and H(-14, 12).
Midpoint = -6 2= ( , )-6 + 12
2( , )6
2 = (-3, 3)
( , )( , )
Example 4:
Find the coordinates of the midpoint of RS for
R(5, -10) and S(3, 6).
Midpoint = 5 + 3 2
82=
-10 + 6 2
-4 2 = (4, -2)
( , )
Example 5: Find the missing endpoint
Find the coordinates of D if E(-6, 4) is the midpoint of DF and F has coordinates (5, -3).
5 + x 2
-3 + y 2
(-6, 4) =
5 + x 2
-3 + y 2
-6 = 4 =
-12 = 5 + x-5 -5
-17 = x
8 = -3 + y+3 +3
11 = y(-17, 11) =
Example 6:
What is the measure of PR if Q is the midpoint of PR?
PQ = QR = 6 – 3x
(6 – 3x) + (6 – 3x) = 14x + 212 – 6x = 14x + 2
+ 6x + 6x12 = 20x + 2-2 - 2
10 = 20x
10 = 20x20 20
½ = x
PR = 14x + 2= 14(½) + 2= 9
Distance Formula – Coordinate Plane to find Distance
The distance d between two points with coordinates (x1, y1) and (x2, y2) is given by
212
212 )()( yyxxd
Based on the Pythagorean Theorem, which we will learn about later.
Example 7: What is the distance?
Find the distance between E(-4, 1) and F(3, -1)
212
212 )()( yyxxd
EF = (-4 – 3)2 + (1 +1)2
EF = 49 + 4
EF = 53
EF ≈ 7.28
Example 8: What is the distance?
Find the distance between S(-2, 14) and R(4, 3)
212
212 )()( yyxxd
SR = (-2 – 4)2 + (14 - 3)2
EF = 36 + 144
EF = 180
EF ≈ 13.416