Lesson 9.5-The Distance Formula

HW:9.5/ 1-14

Isosceles Right ∆Theorem

2x

• 45° – 45° – 90° TriangleIn a 45° – 45° – 90° triangle the hypotenuse is the square root of two * as long as each leg

Theorem

2

• 30° – 60° – 90° TriangleIn a 30° – 60° – 90° triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is the square root of three times as long as the shorter leg

𝑥√3

Problem Solving Strategy

Know the basic triangle rules

Solve for the other sides

Set known information equal to the corresponding part of the basic triangle

New MaterialTHE DISTANCE FORMULA

Coordinate Geometry

Coordinate Geometry - InvestigationUse the Pythagorean Theorem to find the length of the segment

2

4

22 42 c

c 5220 c

47.4c 6

2

22 26 c

10240 c

32.6c

Coordinate Geometry

(AB)2 = (x2 - x1)2 + (y2 - y1)2

The Distance Formula is based on the Pythagorean Theorem

The distance between points A(x1,y1) and B(x2,y2) is given by

Coordinate Geometry - Example

Exploration

• Get your supplies- Graph Paper- ruler- pencil

• Create a large XY coordinate grid

Copy and label these points onto your graph paper, include the coordinates of each point

Exploration

Exploration• Find the distance between the listed attractions• Use the Pythagorean theorem. • Draw right triangle if necessary.

a. Bumper cars to sledge hammera. (-4, -3) to (2, -3)

x

y

Distance = 6

b. Ferris Wheel and Hall of Mirrors(0, 0) and (3, 1)

x

y

3

1

Use the Pythagorean Theorem

=

=10

c

𝑐=√10𝑐 ≈3.16

b. Ferris Wheel and Hall of Mirrors (0, 0) and (3, 1)

22 )10(30 22 )1()3(

16.310

𝑐=√𝑥2+𝑦 2

Using the points and Pythagorean theorem = DISTANCE FORMULA

y Use the Pythagorean theorem

=

= 25

𝑐=√25𝑐=5

c. Refreshment Stand to Ball Toss(-5, 2) to (-2, -2)

x

3

4 c

c. Refreshment Stand to Ball Toss (-5, 2) to (-2, -2)

22 )22(25 22 )4()3(

525

𝑐=√𝑥2+𝑦 2

Using the points and Pythagorean theorem = DISTANCE FORMULA

y Use the Pythagorean theorem

=

=85

𝒄=√𝟖𝟓𝒄≈𝟗 .𝟐𝟐

d. Bumper Cars to Mime Tent(-4, -3) to (3, 3)

x

7

6c

d. Bumper Cars to Mime Tente. (-4, -3) to (3, 3)

22 )33(34

22 )6()7(

22.985

ExplorationIf your car is parked at the coordinates (17, -9),

and each grid unit represents 0.1 mile, how far is from your car to the refreshment stand?

22 29)5(17 d

22 )11(22 d

60.24605 d ≈2.46 Milesunits *0.1 miles

Try to complete this without plotting the location of your car.

Car to Refreshment stand(17, -9) to (-5, 2)

22 02)3(1 d

22 24 d

20d

Find the distance between the points at (1, 2) and (–3, 0).

222

222 yyxxd

47.452

22 63)4(2 d

22 )3(6 d

45d

Find the distance between the points at (2, 3) and (–4, 6).

71.653

Find the distance between the points at (5, 4) and (0, –2).

√ ( 4+2 )2+ (5−0 )2

√ (6 )2+ (5 )2

√36+25

√61≈7.81

Horseshoes Marcy is pitching a horseshoe in her local park. Her first pitch is 9 inches to the left and 3 inches below the pin. What is the distance between the horseshoe and the pin?

&

√¿¿¿√¿¿¿

√90

3√10≈9.49 𝑖𝑛

Homework

Lesson 9.5 - Distance Formula9.5/1-14