10.1 The Distance and Midpoint Formulas

Algebra 2

Mr. Swartz

Objectives/Standard/Assignment

Objectives: 1. Find the distance between two points

in the coordinate plane, and

2. Find the midpoint of a line segment in the coordinate plane.

Geometry Review!

• What is the difference What is the difference between the symbols AB and between the symbols AB and AB?AB?

Segment ABSegment AB

The The lengthlength of of Segment ABSegment AB

The Distance Formula

• The Distance d between the points (x1,y1) and (x2,y2) is :

212

212 )()( yyxxd

The Pythagorean Theorem states that if a right triangle has legs of lengths a and b and a hypotenuse of length c, then a2 + b2 = c2.

Remember!

Find the distance between the two points.

• (-2,5) and (3,-1)(-2,5) and (3,-1)• Let (xLet (x11,y,y11) = (-2,5) and (x) = (-2,5) and (x22,y,y22) = (3,-) = (3,-

1)1)22 )51())2(3( d

3625d

81.761d

Classify the Triangle using the Classify the Triangle using the distance formula (as scalene, distance formula (as scalene, isosceles or equilateral)isosceles or equilateral)

29)61()46( 22 AB

29)13()61( 22 BC

23)63()41( 22 AC

Because AB=BC the triangle is Because AB=BC the triangle is ISOSCELESISOSCELES

C: (1.00, 3.00)

B: (6.00, 1.00)

A: (4.00, 6.00)

C

B

A

The Midpoint Formula

• The midpoint between the two The midpoint between the two points (xpoints (x11,y,y11) and (x) and (x22,y,y22) is:) is:

)2

,2

( 1212 yyxxm

MIDPOINT FORMULA

Find the midpoint of the Find the midpoint of the segment whose endpoints segment whose endpoints

are (6,-2) & (2,-9)are (6,-2) & (2,-9)

2

92,

2

26

2

11,4

Find the coordinates of the midpoint of GH with endpoints G(–4, 3) and H(6, –2).

Substitute.

Write the formula.

Simplify.

Additional Example 1: Finding the Coordinates of a Midpoint

G(–4, 3)

H(6, -2)

Additional Example 2: Finding the Coordinates of an Endpoint

Step 1 Let the coordinates of P equal (x, y).

Step 2 Use the Midpoint Formula.

P is the midpoint of NQ. N has coordinates (–5, 4), and P has coordinates (–1, 3). Find the coordinates of Q.

Additional Example 2 Continued

Multiply both sides by 2.

Isolate the variables.

–2 = –5 + x+5 +5

3 = x

6 = 4 + y

−4 −4

Simplify. 2 = y

Set the coordinates equal.

Step 3 Find the x-coordinate.

Find the y-coordinate.

Simplify.