4/15/2013

1

Unit 6 Lesson 1 – Part I

Connecting Algebra and Geometry

through Coordinates

Slope & Distance

By PresenterMedia.com

AKS 32: Prove simple geometric theorems algebraically using coordinates.

KEY CONCEPTS

Distance on a Number Line

• To find the distance between two points, a and b, on

a number line, find the absolute value of the

difference of a and b. This can be expressed

algebraically as |a – b| or |b – a|.

• For example, to find the distance between –4 and 5,

take the absolute value of the difference of –4 and 5.

• |–4 – 5| = |–9| = 9 or |5 – –4| = |5 + 4| = |9| = 9

• The distance between the numbers –4 and 5 is 9

units.

4/15/2013

2

KEY CONCEPTS

Pythagorean Theorem

• To find the distance between two points on a coordinate

system, we must use the Pythagorean Theorem.

• Right triangles are triangles with one right (90˚) angle.

• The side that is the longest and is always across from the

right angle is called the hypotenuse.

• The two shorter sides are referred to as the legs of the

right triangle.

KEY CONCEPTS - Pythagorean Theorem

• We can use the Pythagorean Theorem to calculate the length of any

one of the three sides.

• For example, to find the length of the hypotenuse of a triangle with

legs of 5 and 7 units, we use the Pythagorean Theorem.

• a2 + b2 = c2 Pythagorean Theorem

• 52+ 72 = c2 Substitute known values.

• 25 + 49 = c2 Simplify.

• 74 = c2 Simplify.

• √74 = √c2 Take the square root of both sides of the

equation.

• c =74 ≈8.6 The length of the hypotenuse of the right

triangle with side lengths 5 and 7 is 74 , or

approximately 8.6 units.

4/15/2013

3

KEY CONCEPTS

Pythagorean Theorem

EXAMPLE 1 Use the Pythagorean Theorem to calculate the distance

between the points (2,5) and (-4, -3).

STEP 1: Plot the points on a coordinate system.

4/15/2013

4

EXAMPLE 1 (cont.)Use the Pythagorean Theorem to calculate the distance between

the points (2,5) and (-4, -3).

STEP 2: Draw lines to form a right triangle, using each

point as the end of the hypotenuse.

EXAMPLE 1 (cont.)Use the Pythagorean Theorem to calculate the distance between

the points (2,5) and (-4, -3).

• STEP 3: Calculate the length of the vertical side, a,

of the right triangle.

• Let (x1, y1) = (2, 5) and (x2, y2) = (-4, -3)

• |y2-y1| = |-3-5| = |-8| = 8

• The length of side a is 8 units.

• STEP 4: Calculate the length of the horizontal side,

b, of the right triangle.

• |x2 – x1| = |–4 – 2| = |–6| = 6

• The length of side b is 6 units.

4/15/2013

5

EXAMPLE 1 (cont.)Use the Pythagorean Theorem to calculate the distance between

the points (2,5) and (-4, -3).

• STEP 5: Use the Pythagorean Theorem to calculate the

length of the hypotenuse, c.

• a2 + b2 = c2 Pythagorean Theorem

• 82 + 62 = c2 Substitute values for a and b.

• 64 + 36 = c2 Simplify each term.

• 100 = c2 Simplify.

• √100 = √c2 Take the square root of both sides

• 10 = c

• The distance between the points (2, 5) and (–4, –3) is 10

units.

EXAMPLE 2Tyler and Arsha have mapped out locations for a game of manhunt. Tyler’s position is

represented by the point (–2, 1). Arsha’s position is represented by the point (–7, 9).

Each unit is equivalent to 100 feet. What is the approximate distance between Tyler

and Arsha?

STEP 1: Plot the points on a coordinate system.

STEP 2: Draw lines to form a right triangle, using each point as the end

of the hypotenuse.

STEP 3: Calculate the length of the vertical side, a, of the right triangle.

STEP 4: Calculate the length of the horizontal side, b, of the right

triangle.

STEP 5: Use the Pythagorean Theorem to calculate the length of the

hypotenuse, c.

4/15/2013

6

EXAMPLE 2 (cont.)Tyler and Arsha have mapped out locations for a game of manhunt. Tyler’s position is

represented by the point (–2, 1). Arsha’s position is represented by the point (–7, 9). Each unit

is equivalent to 100 feet. What is the approximate distance between Tyler and Arsha?

EXAMPLE 3Kevin is standing 2 miles due north of the school. James is standing 4

miles due west of the school. What is the distance between Kevin and

James?