1-1b: 1-1b: The Coordinate Plane- Distance Formula & Pythagorean

Theorem

M(G&M)–10–9 Solves problems on and off the coordinate plane involving distance, midpoint, perpendicular and parallel lines, or slope

GSE:

M(G&M)–10–2 Makes and defends conjectures, constructs geometric arguments, uses geometric properties, or across disciplines or contexts (e.g., Pythagorean Theorem

G-CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

CCSS

Example: Find the measure of the measure of AB..

AA BB

Point A is at 1.5 and B is at 5.

So, AB = 5 - 1 1.5 = 3.55 = 3.5

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?

1. Right angle

2. 2 legs

3. Hypotenuse

Right angle

LEG

Leg – Sides attached to the Right angle

Hypotenuse- Side across from the right angle. Always the longest side of a right triangle.

Pythagorean Formula

222 )()()( hypotenuselegleg

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

10

88.12164

164

10064

108

2

2

222

DC

DC

DC

DC

We got 8 by | -4 – 4|

We got 10 by | 6 - - 4|

Ex. Pythagorean Theorem off the Coordinate Plane

• Find the missing segment- Identify the parts of the triangle 5 in

13 inAns: 5 2 + X 2 = 13 2

Leg 2 + Leg 2 = Hyp 2

hyp

Leg

Leg

25 + X 2 = 169

X 2 = 144

X = 12 in

2) Distance FormulaLets Use the Pythagorean Theorem

2122

12 yyxx

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~

J (-3,5) T (4,2)

d =

x1, y1 x2, y2

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 congruentfor congruent

Assignment