#### Distance Formula

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Distance Formula

1-1b: The Coordinate Plane
- Distance Formula & Pythagorean
Theorem
CCSS
GSE:
M(G&M)–10–9 Solves problems on and off the coordinate plane
involving distance, midpoint, perpendicular and parallel lines, or slope
M(G&M)–10–2 Makes and defends conjectures, constructs geometric
arguments, uses geometric properties, or across disciplines or contexts
(e.g., Pythagorean Theorem
Example: Find the measure of
AB.
A
Point A is at 1.5 and B is at 5.
So, AB = 5 - 1.5 = 3.5
B
Example
• Find the measure of
PR
• Ans: |3-(-4)|=|3+4|=7
• Would it matter if I
asked for the distance
from R to P ?
Ways to find the length of a
segment on the coordinate plane
• 1) Pythagorean Theorem- Can be used on
and off the coordinate plane
•2) Distance Formula – only used on the
coordinate plane
1) Pythagorean Theorem*
* Only can be used with Right Triangles
What are the parts to a RIGHT Triangle?
Hypotenuse- Side across from the
1.
Right angle
right angle. Always the longest
2.
2 legs
side of a right triangle.
3.
Hypotenuse
LEG
Right
angle
Leg – Sides attached to the Right angle
Pythagorean Formula
(leg) (leg) (hypotenuse)
2
2
2
Example of Pyth. Th. on the
Coordinate Plane
Make a right Triangle out of the segment
(either way)
Find the length of each leg of the right Triangle.
Then use the Pythagorean Theorem to find the
Original segment JT (the hypotenuse).
Find the length of CD using the Pythagorean Theorem
We got 10 by | 6 - - 4|
82 102 DC 2
10
64 100 DC 2
164 DC 2
164 DC 12.8
8
We got 8 by | -4 – 4|
Ex. Pythagorean Theorem off the
Coordinate Plane
• Find the missing segment- Identify the
Leg
parts of the triangle
5 in
Leg 2 + Leg 2 = Hyp 2
Ans: 5 2 + X 2 = 13 2
13 in
25 + X 2 = 169
hyp
2
X = 144
X = 12 in
Leg
2) Distance Formula
Lets Use the Pythagorean Theorem
d=
x2 x1 y2 y1
2
2
J (-3,5)
T (4,2)
x1, y1
x2, y2
Identify one as the 1st point and one as
the 2nd. Use the corresponding x and y
values
(4-(-3))2 + (2-(5))2
(4+3)2 + (2-5)2
(7)2 +(-3)2
49+9
=
58
~ 7.6
~
Example of the Distance Formula
• Find the length of
the green segment
Ans: 109 or approximately 10.44
( ) Congruent Segments
• Segments that have the same
length.
If AB & XY have the same length,
Then AB=XY,
but
AB XY
Symbol for congruent
Assignment