10.1 Distance and Midpoint Formulas

1. 10.1 The Distance
and Midpoint
Formulas
Algebra 2
Mr. Swartz

2. Objectives/Standard/Assignment
Objectives:
1.Find the distance between two points
in the coordinate plane, and
2.Find the midpoint of a line segment in
the coordinate plane.

3. Geometry Review!
•What is the difference What is the difference
between the symbols AB and between the symbols AB and
AB?AB?
Segment ABSegment AB
The The lengthlength of of
Segment ABSegment AB

4. The Distance Formula
•The Distance d
between the points
(x
1
,y
1
) and (x
2
,y
2
) is :
2
12
2
12
)()( yyxxd -+-=

5. The Pythagorean Theorem states that if a right triangle has legs
of lengths a and b and a hypotenuse of length c, then a
2
+ b
2
= c
2
.
Remember!

6. Find the distance between
the two points.
• (-2,5) and (3,-1)(-2,5) and (3,-1)
• Let (xLet (x
11,y,y
11) = (-2,5) and (x) = (-2,5) and (x
22,y,y
22) = (3,-1)) = (3,-1)
22
)51())2(3( --+--=d
3625+=d
81.761»=d

7. Classify the Triangle using the Classify the Triangle using the
distance formula (as scalene, distance formula (as scalene,
isosceles or equilateral)isosceles or equilateral)
29)61()46(
22
=-+-=AB
29)13()61(
22
=-+-=BC
23)63()41(
22
=-+-=AC
Because AB=BC the triangle is Because AB=BC the triangle is
ISOSCELESISOSCELES
C: (1.00, 3.00)
B: (6.00, 1.00)
A: (4.00, 6.00)
C
B
A

8. The Midpoint Formula
•The midpoint between the two The midpoint between the two
points (xpoints (x
11,y,y
11) and (x) and (x
22,y,y
22) is:) is:
)
2
,
2
(
1212
yyxx
m
++
=

9. MIDPOINT FORMULA

10. Find the midpoint of the Find the midpoint of the
segment whose endpoints segment whose endpoints
are (6,-2) & (2,-9)are (6,-2) & (2,-9)
÷
ø
ö
ç
è
æ -+-+
2
92
,
2
26
÷
ø
ö
ç
è
æ-
2
11
,4

11. Find the coordinates of the midpoint of GH
with endpoints G(–4, 3) and H(6, –2).
Substitute.
Write the
formula.
Simplify.
Additional Example 1: Finding the Coordinates of a Midpoint
G(–4, 3)
H(6, -2)

12. Additional Example 2:
Finding the Coordinates of an Endpoint
Step 1 Let the coordinates of P equal (x, y).
Step 2 Use the Midpoint Formula.
P is the midpoint of NQ. N has coordinates
(–5, 4), and P has coordinates (–1, 3). Find the
coordinates of Q.

13. Additional Example 2 Continued
Multiply both
sides by 2.
Isolate the
variables.
–2 = –5 + x
+5 +5
3 = x
6 = 4 + y
−4 −4
Simplify. 2 = y
Set the
coordinates equal.
Step 3 Find the x-
coordinate.
Find the
y-coordinate.
Simplify.