Distance, Midpoints & Circles - WordPress.com · The Distance Formula 3 October 8, 2015 22 2 1 2 1...
Transcript of Distance, Midpoints & Circles - WordPress.com · The Distance Formula 3 October 8, 2015 22 2 1 2 1...
Distance,
Midpoints &
Circles Mr. Velazquez
October 8,
2015
Honors Precalculus 1
The
Distance
Formula
October 8, 2015
Honors Precalculus
2
The Distance Formula
October 8, 2015 3
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
The Distance Formula
October 8, 2015 4
Find the distance between 4, −5 and 9,−2 :
The Midpoint Formula
October 8, 2015 5
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) Find the midpoint of the segment
whose endpoints are −1,5 and 6,8 :
−1 + 6
2,5 + 8
2
5
2,13
2
The Midpoint Formula
October 8, 2015 6
Find the midpoint of the segment whose endpoints are −1,7
and −5, 9 :
Classwork/Exit Ticket
October 8, 2015 7
A line segment, 𝐿 , extends from the point
−3,−1 to the point 7, 5 . Use the
distance and midpoint formulas to find
the length of 𝐿 and the coordinates of
the midpoint of 𝐿 .
Circles
October 8, 2015 8
Graphing Circles
October 8, 2015 9
Since the equation representation of a circle is not a function, to
graph a circle using a graphing utility, we must consider two
separate functions when solving for y:
𝑥2 + 𝑦2 = 4
𝑦2 = 4 − 𝑥2
𝑦 = ± 4 − 𝑥2
𝑦 = + 4 − 𝑥2 and 𝑦 = − 4 − 𝑥2
So we graph both the positive and negative versions of y
separately, which together will form a circle.
Circles
October 8, 2015 10
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
October 8, 2015 11
Find the center and radius of the circle whose
equation is 𝑥 + 3 2 + 𝑦 − 4 2 = 9
Graph the equation.
Use the graph to identify the
relation’s domain and range. Why is
it a relation and not a function?
x
y
3
(-3,4)
Center: (−3, 4) Radius: 3
Domain: [−6, 0]; Range: [1, 7]
Circles
October 8, 2015 12
Write the standard form of the equation of the circle with center
(−2, 7) and radius of 5.
Circles
October 8, 2015 13
Find the center and radius of the circle represented by the equation
𝑥 − 6 2 + 𝑦 + 5 2 = 49. Find the domain and range and graph it.
x
y
Circles, General Form
October 8, 2015 14
Using the previous equation, we can multiply out the factors
and move all terms to the left side, combining like terms to
obtain the general form of the equation.
𝑥 − 6 2 + 𝑦 + 5 2 = 49
𝑥2 − 12𝑥 + 36 + 𝑦2 + 10𝑦 + 25 = 49
𝑥2 + 𝑦2 − 12𝑥 + 10𝑦 + 12 = 0
General Form
Circles, General Form
October 8, 2015 15
Complete the square and write the equation in standard form.
Then give the center and radius of the circle.
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
Example
October 8, 2015 16
Complete the square and write the equation in standard form. Then
give the center and radius of the circle and graph the equation.
x
y 2 2x 4 12 15 0y x y
Example
October 8, 2015 17
Complete the square and write the equation in standard form. Then
give the center and radius of the circle and graph the equation.
x
y2 2x 6 8 0y x y
Classwork/Exit Ticket
October 8, 2015 18
You are trying to draw the graph of a dartboard on a
coordinate plane. The design of this dartboard is simple: a total
of four concentric (same center) circles, where the radius of the
outer circle is 1 unit longer than the radius of the next largest
circle, whose radius is 1 unit longer than the next, etc. The area
of the entire dartboard (i.e. the outer circle) should be at least
25𝜋, and the center of the dartboard should be located
somewhere in the first quadrant.
Choose a center and outer radius that fits the requirements,
and write four equations (in standard form) to represent each of
the four circles needed to draw your dartboard. Then graph
your dartboard on a sheet of graph paper.