Pythagorean Theorem
In a right triangle, the sum of the squares of the two
shorter sides is equal to the square of the hypotenuse.
a2  b2  c2
leg 2  leg 2  hyp 2
c
a
(leg)
(hyp)
b
(leg)
(“c” is always the hypotenuse)
Distance Formula
d
x2  x1 2   y2  y1 2
Where d stands for distance
x1 & y1 are one endpoint of a segment
x2 & y2 are the second endpoint of a
segment
(x2 , y2)
(x1 , y1)
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 AB , then
AX  XB
(the measure of AX = the measure of XB)
B
X

A
Midpoint Formula
On a Single Number Line:
X
A B
2
5
Units
On a Double Number Line
(Coordinate Plane):
x
x1  x2
y  y2
& y 1
2
2
y
5 Units
(x2,y2)
(5,5)
-3
2
7
A
X
B
(1,2
)

x
(-3,-
(x1)
1,y1)
3 7 4
 2
2
2
x
35
2
 22  1
y
15
2
 42  2
M
(a single number line)
ab
M 
2
 20  40
M
2
20
M
 10
2
QM  MP
(a double number line)
x1  x2
x
&
2
6   1
x
&
2
5
x   2.5 &
2
M(2.5, 1.5)
y1  y2
y
2
1 2
y
2
3
y   1.5
2
 P(-1, 2)
 M(2.5, 1.5)
 Q(6, 1)
(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.
(a single number line)
A B
M
2
 12  B
2
2  5
2
10  12  B
 12  12
22  B
-12

A
22
?
5
17 units

M
17 units
Multiply both sides by 2
Solve for B

B
Let X = (x1,y1) (One endpoint)
(x, y)
Y = (-2, 2)
(x2, y2)
Z = (2, 8)
(Midpoint)
(The midpoint is always
the ordered pair with
no subscripts)

Y(-2,2)
(-4, -6)
(x, y)
X(?)
(Other endpoint)(x1, y1)
x1  x2
2
x 2
2
2  2 1
2
x
y1  y2
2
y 8
2 1
2
2
y
2 
 4  x1  2
4  y1  8
 6  x1
 4  y1
Z(2, 8)
(x2, y2)
Look for an equation to
write, then solve:
4x  5  11  2x
-2x
-2x
2x  5  11
+5 +5
2x  16
x 8
Be sure to answer the question.
BC  11  2 x
 11  28  27
Does it work?
4x  5  11  2x
?
48  5  11  28
?
32  5  11  16
 27  27
Freebies
HW: Page 25 (3 – 45 odds)