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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: 1.9 – Pg. 250, #2-60 (evens)