Distance, Midpoints, and Circles - Battaly · 2.8 Distance, Midpoints, and Circles Class Notes:...
Transcript of Distance, Midpoints, and Circles - Battaly · 2.8 Distance, Midpoints, and Circles Class Notes:...
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Distance, Midpoints, and Circles
© G. Battaly 2015 1
November 22, 2015
2.8 Distance, Midpoints, and Circles
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Study 2.8 CVC # 15; # 119; 23, 27, 31, 35, 39 41, 45, 49, 53, 57, 61
College Algebra & Trig Home Page Homework
GOALS:1. Understand the distance formula.2. Find the distance between 2 points.3. Find the midpoints of the line segment joining 2 points.4. Recognize the general and standard forms of the equation for a circle.5. Identify the center and radius of a circle from the equation in standard form.6. Find the center and radius of a circle by converting from general form to standard form.
2.8 Distance, Midpoints, and Circles
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Pythagorean Theorum
a2 + b2 = c2
College Algebra & Trig Home Page Homework
ac
b
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Distance, Midpoints, and Circles
© G. Battaly 2015 2
November 22, 2015
2.8 Distance, Midpoints, and Circles
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Pythagorean Theorum
a2 + b2 = c2
College Algebra & Trig Home Page Homework
ac
b
To find c, knowing a and b, need to solve for c:
c2 = a2 + b2
c = √ a2 + b2
2.8 Distance, Midpoints, and Circles
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Pythagorean Theorum
a2 + b2 = c2
College Algebra & Trig Home Page Homework
ac
b
(x1,y1)
(x2,y2)
To find c, knowing a and b, need to solve for c:
c2 = a2 + b2
c = √ a2 + b2 b = x2 x1
a = y2 y1then:c = √ (x2 x1)2 + (y2 y1)2
Distance from (x1,y1) to (x2,y2)
d = √ (x2 x1)2 + (y2 y1)2
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Distance, Midpoints, and Circles
© G. Battaly 2015 3
November 22, 2015
2.8 Distance, Midpoints, and Circles
Class Notes: Prof. G. Battaly, Westchester Community College, NY
College Algebra & Trig Home Page Homework
Distance from (x1,y1) to (x2,y2)
d = √ (x2 x1)2 + (y2 y1)2
G: (5,1) and (8,5) F: distance between pts.
2 4 6 8
8
6
4
2
d = √ (x2 x1)2 + (y2 y1)2
d = √ ( )2 + ( )2
2 4 6 8
8
6
4
2
2.8 Distance, Midpoints, and Circles
Class Notes: Prof. G. Battaly, Westchester Community College, NY
College Algebra & Trig Home Page Homework
Distance from (x1,y1) to (x2,y2)
d = √ (x2 x1)2 + (y2 y1)2
G: (5,1) and (8,5) F: distance between pts.
d = √ (x2 x1)2 + (y2 y1)2
d = √ ( )2 + ( )2
d = √ ( 8 5)2 + ( 5 1)2
d = √ ( 3)2 + ( 4)2
= √ 25 = 5
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Distance, Midpoints, and Circles
© G. Battaly 2015 4
November 22, 2015
2.8 Distance, Midpoints, and Circles
Class Notes: Prof. G. Battaly, Westchester Community College, NY
College Algebra & Trig Home Page Homework
G: (4,1) and (2,3) F: distance between pts.
d = √ (x2 x1)2 + (y2 y1)2
d = √ ( )2 + ( )2
d = √ (x2 x1)2 + (y2 y1)2
(4,1)
(2,3)
d
2.8 Distance, Midpoints, and Circles
Class Notes: Prof. G. Battaly, Westchester Community College, NY
College Algebra & Trig Home Page Homework
G: (4,1) and (2,3) F: distance between pts.
d = √ (x2 x1)2 + (y2 y1)2
d = √ ( )2 + ( )2
d = √ [2 (4)]2 + [3 (1)]2
d = √ ( 6)2 + (2)2
= √ 40 = 2√ 10
d = √ (x2 x1)2 + (y2 y1)2
(4,1)
(2,3)
d
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Distance, Midpoints, and Circles
© G. Battaly 2015 5
November 22, 2015
ac
b(x1,y1)
(x2,y2)
2.8 Distance, Midpoints, and Circles
Class Notes: Prof. G. Battaly, Westchester Community College, NY
College Algebra & Trig Home Page Homework
Find the midpoint of line b:
half the distance = x2 x1 2add to x1: x2 x1 + 2x1 2 2midpoint at x = x1 + x2 2
ac
b(x1,y1)
(x2,y2)
2.8 Distance, Midpoints, and Circles
Class Notes: Prof. G. Battaly, Westchester Community College, NY
College Algebra & Trig Home Page Homework
Find the midpoint of line a: half the distance = y2 y1 2add to y1: y2 y1 + 2y1 2 2midpoint at y = y1 + y2 2
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Distance, Midpoints, and Circles
© G. Battaly 2015 6
November 22, 2015
ac
b(x1,y1)
(x2,y2)
2.8 Distance, Midpoints, and Circles
Class Notes: Prof. G. Battaly, Westchester Community College, NY
College Algebra & Trig Home Page Homework
Find the midpoint of line c: midpoint at:
x1 + x2 , y1 + y2 2 2
2.8 Distance, Midpoints, and Circles
Class Notes: Prof. G. Battaly, Westchester Community College, NY
College Algebra & Trig Home Page Homework
G: (10,4) and (2,6) F: midpoint of line segment joining pts.
x1 + x2 , y1 + y2 2 2
(6,5)
x1 + x2 , y1 + y2 2 2
(10,4)
(2,6)
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Distance, Midpoints, and Circles
© G. Battaly 2015 7
November 22, 2015
2.8 Distance, Midpoints, and Circles
Class Notes: Prof. G. Battaly, Westchester Community College, NY
College Algebra & Trig Home Page Homework
G: (10,4) and (2,6) F: midpoint of line segment joining pts.
x1 + x2 , y1 + y2 2 2
(6,5)
x1 + x2 , y1 + y2 2 2
10 + 2 , 4 + 6 2 2
12 , 10 2 2
6 , 5
(2,6)
(10,4)
2.8 Distance, Midpoints, and Circles
Class Notes: Prof. G. Battaly, Westchester Community College, NY
College Algebra & Trig Home Page Homework
G: (2,1) and (8,6) F: midpoint of line segment joining pts.
x1 + x2 , y1 + y2 2 2
(2,1)
(8,6)
x1 + x2 , y1 + y2 2 2
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Distance, Midpoints, and Circles
© G. Battaly 2015 8
November 22, 2015
2.8 Distance, Midpoints, and Circles
Class Notes: Prof. G. Battaly, Westchester Community College, NY
College Algebra & Trig Home Page Homework
G: (2,1) and (8,6) F: midpoint of line segment joining pts.
x1 + x2 , y1 + y2 2 2
(2,1)
(8,6)
x1 + x2 , y1 + y2 2 2
(2) + (8) , (1) + 6 2 2
10 , 5 2 2
5 , 5 2
2.8 Distance, Midpoints, and Circles
Class Notes: Prof. G. Battaly, Westchester Community College, NY
College Algebra & Trig Home Page Homework
Circle: Set of all Points Equidistant from a fixed point. The fixed point is the Center. The fixed distance is the Radius
Cr
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Distance, Midpoints, and Circles
© G. Battaly 2015 9
November 22, 2015
2.8 Distance, Midpoints, and Circles
Class Notes: Prof. G. Battaly, Westchester Community College, NY
College Algebra & Trig Home Page Homework
Circle: Standard Form: (xh)2 + ( yk)2 = r2
C (h,k)r (x,y)
2.8 Distance, Midpoints, and Circles
Class Notes: Prof. G. Battaly, Westchester Community College, NY
College Algebra & Trig Home Page Homework
Circle: General Form: x2 + y2 + Dx + Ey + F = 0
C (h,k)r (x,y)
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Distance, Midpoints, and Circles
© G. Battaly 2015 10
November 22, 2015
2.8 Distance, Midpoints, and Circles
Class Notes: Prof. G. Battaly, Westchester Community College, NY
College Algebra & Trig Home Page Homework
Circle: x2 + y2 = 25
(xh)2 + ( yk)2 = r2
∴ C(0,0) and r = 5
2.8 Distance, Midpoints, and Circles
Class Notes: Prof. G. Battaly, Westchester Community College, NY
College Algebra & Trig Home Page Homework
Circle: x2 + y2 = 25
(xh)2 + ( yk)2 = r2
∴ C(0,0) and r = 5
C(0,0)
r = 5
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Distance, Midpoints, and Circles
© G. Battaly 2015 11
November 22, 2015
2.8 Distance, Midpoints, and Circles
Class Notes: Prof. G. Battaly, Westchester Community College, NY
College Algebra & Trig Home Page Homework
(xh)2 + ( yk)2 = r2
∴ C( __, __ ) and r = __
G: (x2)2 + (y1)2 = 16 F: C, r, sketch
2.8 Distance, Midpoints, and Circles
Class Notes: Prof. G. Battaly, Westchester Community College, NY
College Algebra & Trig Home Page Homework
(xh)2 + ( yk)2 = r2
∴ C(2,1) and r = 4
C(2,1)
r = 4
G: (x2)2 + (y1)2 = 16 F: C, r, sketch
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Distance, Midpoints, and Circles
© G. Battaly 2015 12
November 22, 2015
2.8 Distance, Midpoints, and Circles
Class Notes: Prof. G. Battaly, Westchester Community College, NY
College Algebra & Trig Home Page Homework
G: x2 + y2 + 4x + 2y + 1 = 0 F: C, r, sketch
x2 + y2 + Dx + Ey + F = 0
x2 + 4x + __ + y2 + 2y + __ = 1 + __
2.8 Distance, Midpoints, and Circles
Class Notes: Prof. G. Battaly, Westchester Community College, NY
College Algebra & Trig Home Page Homework
x2 + y2 + Dx + Ey + F = 0
x2 + 4x + _4_ + y2 + 2y + _1_ = 1 + _5_
(x + _)2+ (y + _)2 = _
G: x2 + y2 + 4x + 2y + 1 = 0 F: C, r, sketch
∴ C(__,__), r = __
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Distance, Midpoints, and Circles
© G. Battaly 2015 13
November 22, 2015
2.8 Distance, Midpoints, and Circles
Class Notes: Prof. G. Battaly, Westchester Community College, NY
College Algebra & Trig Home Page Homework
x2 + y2 + Dx + Ey + F = 0
x2 + 4x + _4_ + y2 + 2y + _1_ = 1 + _5_
(x + 2)2+ (y + 1)2 = 4
G: x2 + y2 + 4x + 2y + 1 = 0 F: C, r, sketch
∴ C(2,1), r = 2
C(2,1)r = 2
2
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