Download - 1.3 Distance & Midpoint p. 21
Pythagorean Theorem
222 hyplegleg
In a right triangle, the sum of the squares of the two shorter sides is equal to the square of the hypotenuse.
222 cba (“c” is always the hypotenuse)
a
b
c
(leg)
(leg)
(hyp)
Distance Formula
212
212 yyxxd
Where d stands for distance
x1 & y1 are one endpoint of a segment
x2 & y2 are the second endpoint of a segment
(x1 , y1)
(x2 , y2)
d
What do you say when you walk into a cold room?
It’s collinear!
(It’s cold in here!)
Midpoint of a Segment
The point halfway between the endpoints of a segment
If X is the midpoint of Segment AB , then
ABXBAX
A
B
X
(the measure of AX = the measure of XB)
Midpoint Formula
On a Single Number Line:
On a Double Number Line (Coordinate
Plane):2
BAX
2 &
22121 yy
yxx
x
A B
-3 7
22
4
2
73
X
2
5 Units
5 Units
x
y(5,5)
(-3,-1)(x1,y1
)
(x2,y2
)
122
253 x 22
42
51 y
(1,2)
2
4020 M
2
baM
(a single number line)
102
20M
M
MPQM
2 &
22121 yy
yxx
x
(a double number line)
2
21 &
2
16
yx
5.12
3 & 2.5
2
5 yx
M(2.5, 1.5)
2) P(-1,
1) Q(6,
1.5) M(2.5,
(Not to scale)
Find the Coordinates of an Endpoint
The Midpoint Formula can be used to find the coordinates of an endpoint when the midpoint and one endpoint are given.Find the coordinates of Endpoint B, if M = 5 is the midpoint, and A = -12 is the other endpoint.
2
BAM
2
125
B Multiply both sides by 22 2
B 1210 Solve for B12 12
B22
A
-12 M
5 B
?22
17 units 17 units
(a single number line)
Let X = (x1,y1) (One endpoint)
Y = (-2, 2) (Midpoint)
Z = (2, 8) (Other endpoint)(x2, y2)
(x, y)
X(?)
Y(-2,2)
Z(2, 8)
(x, y)
(x2, y2)
(x1, y1)
(The midpoint is always the ordered
pair with no subscripts)
221 xx
x
2
22 1 x
2 2
24 1 x
16 x
2
82 1 y
221 yy
y
2 2
84 1 y
14 y
(-4, -6)
Look for an equation to write, then solve:
xx 21154 -2x-2x
1152 x
162 x+5+5
8x
Does it work?xx 21154
8211584 ?
1611532 ?
2727 Be sure to answer the question.
278211
211
xBC
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