Equations of Circles 4.1 Equations of circles Arcs, Inscribed Angles, Central Angles

What you’ll learn about Circles Circle equation Coordinates of the center Measure of the radius Inscribed and Central Angles Arcs … and why Circles are cool! Random thought…. How many sides does a circle have? (two, inside and outside )

Equation of a Circle The center of a circle is given by (h, k) The radius of a circle is given by r The equation of a circle in standard form is (x – h) 2 + (y – k) 2 = r 2

Finding the Equation of a Circle Circle A The center is (16, 10) The radius is 10 The equation is (x – 16) 2 + (y – 10) 2 = 100

Finding the Equation of a Circle Circle B The center is (4, 20) The radius is 10 The equation is (x – 4) 2 + (y – 20) 2 = 100

Finding the Equation of a Circle Circle O The center is (0, 0) The radius is 12 The equation is x 2 + y 2 = 144

Graphing Circles (x – 3) 2 + (y – 2) 2 = 9 Center (3, 2) Radius of 3

Graphing Circles (x + 4) 2 + (y – 1) 2 = 25 Center (-4, 1) Radius of 5

Graphing Circles (x – 5) 2 + y 2 = 36 Center (5, 0) Radius of 6

Writing Equations of Circles Write the standard equation of the circle: Center (4, 7) Radius of 5 (x – 4) 2 + (y – 7) 2 = 25

Writing Equations of Circles Write the standard equation of the circle: Center (-3, 8) Radius of 6.2 (x + 3) 2 + (y – 8) 2 = 38.44

Writing Equations of Circles Write the standard equation of the circle: Center (2, -9) Radius of (x – 2) 2 + (y + 9) 2 = 11

Writing Equations of Circles Write the standard equation of the circle: Center (0, 6) Radius of x 2 + (y – 6) 2 = 7

Writing Equations of Circles Write the standard equation of the circle: Center (-1.9, 8.7) Radius of 3 (x + 1.9) 2 + (y – 8.7) 2 = 9

Circles Circles Why 360 º ? The idea of dividing a circle into 360 equal pieces dates back to the sexagesimal (60-based) counting system of the ancient Sumarians. Early astronomical calculations linked the sexagesimal system to circles.

Angles vs. lengths If I lengthen the sides of the angle, does the measure of the angle change? NO A Notice that the length of the arc depends upon how far the arc is from the vertex of the angle. 37°

Angles “inscribed” on a circle. Inscribed angle: has its vertex on the circle and sides that intersect the circle at point other than the vertex.

“Central” Angle of a circle. Central angle: has its vertex at the center of the circle.

What does “interior of an angle” mean? Which picture illustrates the “interior of an angle”? “interior of an angle” “exterior of an angle”

“Intercepted” Arc Intercepted arc: the arc of the circle that is in the interior of the angle.

There is a relationship between the measure of the “Central” Angle and the measure of the “inscribed” angle that intercepts the same arc. Relationship Central Angle Theorem

The measure of a central angle is twice that of an inscribed angle that intercepts the same arc. The “Inscribed/Central angle” theorem

Arc measure = central angle measure “The Pac-Man” Arc measure = 2*inscribed angle measure “The Pistachio”

For circle Z

Arc BD could be the short arc OR the long arc. We would call the “short” arc the minor arc BD or arc BZD

Arc BD could be the short arc OR the long arc. We would call the “long” arc the major arc BD or arc BAD

The central angle that subtends the entire circle has a measure of 360 degrees. m(arc BZ) = m(arc ZD) + m(arc DA)+ m(arc AB)

Find the measure of the angle.