Presentation on theme: "Circles > Formulas Assignment is Due. Center Circle: A circle is the set of all points in a plane that are equidistant from a given point called the center."— Presentation transcript:
Center Circle: A circle is the set of all points in a plane that are equidistant from a given point called the center of the circle. A circle with center P is called “ circle P ” or P Radius Diameter Chord Tangent Secant
Standard Equation of a Circle r 2 = (x-h) 2 + (y-k) 2 Where, r = radius (h,K) = center of the circle
Example: Write the standard equation of a circle with center (2,-1) and radius = 2 r 2 = (x-h) 2 + (y-k) 2 2 2 = (x- 2) 2 + (y- -1) 2 4 = (x-2) 2 + (y+1) 2
Example: Give the coordinates for the center, the radius and the equation of the circle Center: Radius: Equation: Center: Radius: Equation: (-2,0) 4 4 2 =(x-(-2)) 2 +(y-0) 2 (0,2) 2 2 2 =(x-0) 2 +(y-2) 2 16=(x+2) 2 +y 2 4=x 2 +(y-2) 2
Rewrite the equation of the circle in standard form and determine its center and radius x 2 +6x+9+y 2 +10y+25=4 (x+3) 2 (y+5) 2 + =2 2 Center: (-3,-5) Radius: 2
Rewrite the equation of the circle in standard form and determine its center and radius x 2 -14x+49+y 2 +12y+36=81 (x-7) 2 (y+6) 2 + =9 2 Center: (7,-6) Radius: 9
Use the given equations of a circle and a line to determine whether the line is a tangent or a secant Circle: (x-4) 2 + (y-3) 2 = 9 Line: y=-3x+6
Example: The diagram shows the layout of the streets on Mexcaltitlan Island. 1. Name 2 secants 2. Name two chords 3. Is the diameter of the circle greater than HC? 4. If ΔLJK were drawn, one of its sides would be tangent to the circle. Which side is it?
THM: If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency. P l Q If l is tangent to circle Q at P, then
If BC is tangent to circle A, find the radius of the circle. Use the pyth. Thm. r 2 +24 2 = (r+16) 2 r 2 +576 = (r+16)(r+16) r 2 +576 = r 2 +16r+16r+256 r 2 +576 = r 2 +32r+256 -r 2 576 = 32r + 256 -256 320 = 32r 32 10 = r A 16 24 r r BC
Example: A green on a golf course is in the shape of a circle. A golf ball is 8 feet from the edge of the green and 28 feet from a point of tangency on the green, as shown at the right. Assume that the green is flat. 1. What is the radius of the green 2. How far is the golf ball from the cup at the center?
Thm: If 2 segments from the same exterior point are tangent to a circle, then they are congruent. R T S P If SR and TS are tangent to circle P, then
AB and DA are tangent to circle C. Solve for x. X 2 – 7x+20 = 8 X 2 7x+12= 0 (x-3)(x-4)=0 X=3, x=4 B D C A X 2 -7x+20 8
Inside Angles – if two chords intersect in the interior of a circle, then the measure of each angle is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle 1 A B D C m<1 = ½( mDC + mAB)
Example: Find the missing angle 40 20 A B C D 1 m<1 = ½( mDC + mAB) m<1 = ½( 40+20) m<1 = ½(60) m<1 = 30
Outside Angles 0 If a tangent and a secant, two tangents, or two secants intersect in the exterior of a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs. 1 A B C m<1 = ½( mAB - mBC)
Example: find the missing angle X = ½ (264-96) X = ½ (168) X=84 96 X 264
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