Day 72: Circles

1. Day 72:Circles Getting dizzy yet?

2. Recap • Radius • Diameter • Circumference • Pi • Area • Central Angle • Arc • Arc Measure • Arc Length • Sector • Area of a Sector • Inscribed Angle • Semicircle • Tangent/Radius • Diameter/Chord • Intersecting Chords • Intersecting Secants • Tangent/Secant • Tangent/Tangent

3. Circle Constructions • Constructing a circle with a given radius • Copy the radius with your compass, place the point down, and spin it all the way around. • Finding a circle’s center. • This relies on the fact that a radius bisects a chord if and only if they are perpendicular. (See video) • Constructing a circle given three points. • Any three non-collinear points are concyclic (lie on the same circle), and a unique circle can be constructed through them. We are basically going to find the circumcenter of the triangle formed by the points. (See video) • Tangent to a point on the circle. • Remember a tangent is always perpendicular to a radius, so we’ll construct the perpendicular bisector to a segment that contains the radius. (See video) • Tangent(s) to a point outside the circle. • There will be two of these, and they are congruent. • Challenge problem: look up constructing an inscribed hexagon. • Extra-challenge problem: look up constructing an inscribed pentagon

4. Equation of a Circle • A circle can be represented algebraically (as an equation of two variables). • For any circle with its center at the origin ((0,0) on the coordinate plane), its equation is: x2 + y2 = r2 where r is the radius.

5. x2 + y2 = r2 • Does this formula seem familiar to anyone? • Something squared plus something squared equals something squared? • The formula for a circle is based on the Pythagorean Theorem.

6. Circles and the Pythagorean Theorem • Let’s pick some radius r. • To find the coordinates of the endpoint, count x over and y up. • So by the Pythagorean Theorem… • x2 + y2 = r2 • What if the radius pointed in a different direction? • We get the same equation. • What if we considered all of the radii? • So the equation represents all of the (x, y) values that are a given distance away from the center, which is also the definition of a circle. r x y x y

7. Graphing a Circle • What is the radius for the following circle? x2 + y2 = 9 • Plot 4 points based on the radius and draw the circle. • The circle represents all the values for x and y that make the equation true.

8. Example • What is the equation of this circle? • x2 + y2 = 4

9. Shifting the Circle • Not all graphed circles have their centers at the origin. • The general equationof a circle with center (h, k) is (x – h)2 + (y – k)2 = r2 • Note: The center has the opposite sign of h and k. • Answer: (x – 1)2 + y2 = 1

10. Example • What is the equation for this circle? • Find the center andradius. • Plug the center intothe equation withthe signs switched. • Plug in the radius. • (x – 2)2 + (y + 3)2 = 16

11. Example • Graph the following circle: (x – 3)2 + (y – 1)2 = 4 • Plot the center. • Use the radius toplot 4 points.

12. Find the center and radius of the circle

13. Find the Equation of the Circle

14. Find the Equation of the Circle Find the equation of a circle with: • center at (1, 2) and a point on the circle is (4, 3) • center at (3, -1) and a point on the circle is (5, 5) • a diameter with end points of (-4, 6) and (6, 2)

15. Find the Equation of the Circle • Find the equation of a circle that passes through A(1, 2), B(-3, 4), and C(-5, 0). • Remember that the perpendicular bisectors of chords will meet at the center. • The center will be equidistant from all three points.

16. Homework 45 • Workbook, p. 137 • Handout: Circles & Tangents