midpoint and distance formulas...the midpoint formula find the midpoint of the segment whose...

MIDPOINT AND DISTANCE FORMULAS Mr. Velazquez

Honors Precalculus

Suppose we wanted to obtain a formula for the distance 𝑑, assuming we happen to know the coordinates of the two points 𝑃1 and 𝑃2

THE DISTANCE FORMULA

( ) ( )

( ) ( )

2 2

2 1 2 1

2 2

Find the Distance between (-4,2) and (3,-7)

x

3 4 7 2

49 81

130 11.4

x y y− + −

− − + − −

+

=

− − − − − − −

−

−

−

−

−

−

−

−

x

y

EXAMPLES:

Find the distance between (4,-5) and (9,-2).

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

THE MIDPOINT FORMULA

Find the midpoint of the segment

whose endpoint are (-1,5) and (6,8).

1 6 5 8,

2 2

5 13,

2 2

− + +

−

−

−

x

y

Find the midpoint of the segment

whose endpoint are (-1,5) and (6,8).

1 6 5 8,

2 2

5 13,

2 2

− + +

(-1,5)

(6,8)

EXAMPLES:

Find the midpoint of the segment whose endpoints

are (-1,7) and (-5,9).

The midpoint of a segment is at (2,-1) and one of

its endpoints is at (-6,8). Find the coordinates of

the other endpoint.

EXAMPLES:

A line segment ത𝐿 extends from the

point −3,−1 to the point 7, 5 . Find

the length of ത𝐿 and the coordinates of

the midpoint of ത𝐿.

CIRCLES

GRAPHING CALCULATORS

2 2

2 2

2

To Graph a Circle;

First Solve the equation for y: x 4

y 4-x

y = 4

Graph as t

y

x

+ =

=

−

2 2

1 2wo separate equations y = 4 y = 4

So that the circle doesn't look flattened, press ZOOM, #5 for ZSquare.

Now press GRAPH.

x x− − −

CIRCLES

Write the standard form of the

equation of the circle with center

(-4,1) and radius of 3.

− − − − − − −

−

−

−

−

−

−

−

−

x

y

2 2 2

2 2

( 4) ( 1) 3

( 4) ( 1) 9

x y

x y

−− + − =

+ + − =Standard

Form

(-4,1)

3

CIRCLES

Find the center and radius of the

circle whose equation is

Graph the equation.

Use the graph to identify the

relation’s domain and range. Why is it

a relation and not a function?

2 2( 3) ( 4) 9x y+ + − =

− − − − − − −

−

−

−

−

−

−

−

−

x

y

3

(-3,4)

Domain: [-6,0]; Range:[1,7]

Center(-3,4); radius=3

EXAMPLES:

Write the standard form of the equation of the

circle with center at (-2,7) and a radius of 5.

Find the center and radius of the circle whose equation is

below. Graph the equation. Use the graph to identify the

relation’s domain and range. 𝑥 − 1 2 + 𝑦 − 2 2 = 16

GENERAL FORM OF A CIRCLE

2 2

2 2

If we take the equation from the previous

problem we can multiply out the factors

and move all terms to one side to get the

general form of the equation of the circle.

( 6) ( 5) 49

12 36 10 25

x y

x x y y

− + + =

− + + + +

2 2

49

12 10 12 0x y x y

=

+ − + + = General Form

COMPLETING THE SQUARE

When given the equation of a circle in general form, we can use completing the square to return it to standard form and find the center and radius.

𝑥2 + 𝑦2 − 14𝑥 + 8𝑦 + 29 = 0

𝑥2 − 14𝑥+ ? +𝑦2 + 8𝑦+ ?= −29

(𝑥2 − 14𝑥 + 𝟒𝟗) + (𝑦2 + 8𝑦 + 𝟏𝟔) = −29 + 𝟒𝟗 + 𝟏𝟔

𝑥 − 7 2 + 𝑦 + 4 2 = 36

Center: (7, −4) Radius: 6

EXAMPLES:

Complete the square for each of the following circles

and write the equations in standard form. Then give

the center and radius of each circle.2 2x 4 12 15 0y x y+ − + + =

2 2x 6 8 0y x y+ + − =

CLASSWORK & HOMEWORK

MATH JOURNAL: Summarize what you learned today

NO CLASSWORK FOR THIS CHAPTER (except your math journal)

HOMEWORK: Functions HW 4 (Math XL)