Notes for distance and midpoint

1. Section 1-3
Jim Smith JCHS
Spi.2.1.E

2. The distance between 2 points is
the absolute value of the difference
of the coordinates.
The distance between
exit 417 and 407 is
| 417 – 407 | = 10 or
| 407 – 417 | = | -10 | = 10

3. The distance between A and B is
| | | | | | | | | | | | | |
-5 4
A B
| -5 – 4 | = | -9 | = 9
Our distances should always
be positive

4. | | | | | | | | | | | | | |
-6 12
A B
The midpoint of a segment
is the average of the
coordinates
-6 + 12
2
= 6
2
= 3

5. Review Graphing
y
x
( 0,0 )
Origin
Positive
Negative
Order ( X,Y )

6. Graphing
y
x

8. A
B
The Distance Formula Is Derived
From The Pythagorean Formula
6
15
6² + 15² = AB²
√261

9. Distance Formula
Dist = ( x - x )² + ( y - y )²
Remember the order ( x , y )
Check yourself …
our answers should be positive

10. Find the distance between:
( 3 – 8 )² + ( 6 - 10 )²
( -5 )² + ( -4 )²
25 + 16
41 = 6.40
( 8 – 3 )² + ( 10 – 6 )²
( 5 )² + ( 4 )²
25 + 16
41 =6.40
( 3 , 6 ) and ( 8 , 10 )

11. MIDPOINT
The midpoint of a segment is half way
between the x’s and half way between the y’s
You can call it the average
6
10
Midpoint

12. Midpoint Formula
X + X , Y + Y
2 2
Find the midpoint of
( 2,8 ) and ( 6,4 )
2 + 6 , 8 + 4 = 8 ,12 = ( 4 , 6 )
2 2 2 2

13. X1 + X2
2
= XMID
Y1 + Y2
2
= YMID
What If We Knew The Midpoint Of A Segment
And One Endpoint? How Would We Find The
Other Endpoint?
Think Of The Formula As:
Endpoints Midpoint
( X1 , Y1 ) ( X2 , Y2 ) ( Xmid , Ymid )

14. Endpoint ( 3 , 5 ) Midpoint ( 6 , -2 )
Find The Other Endpoint.
X1 + X2
2
= XMID
Y1 + Y2
2
= YMID
Find ( X2 ,Y2 )
3 + X2
2
3 + X2
X2 = 9
= 6
= 12
5 + Y2
2
5 + Y2
Y2 = -9
= -2
= -4
( 9 , -9 )