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Formulas

Formulas. Things you should know at this point. Measure of an Inscribed Angle. If an angle is inscribed in a circle, then its measure is one half the measure of its intercepted arc.  = . Theorem. The measure of an  formed by 2 lines that intersect inside a circle is.

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Formulas

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  1. Formulas Things you should know at this point

  2. Measure of an Inscribed Angle • If an angle is inscribed in a circle, then its measure is one half the measure of its intercepted arc.  =

  3. Theorem. The measure of an  formed by 2 lines that intersect insidea circle is Measure of intercepted arcs

  4. Theorem. The measure of an  formed by 2 lines that intersect outsidea circle is Smaller Arc 3 cases: Larger Arc 2 Secants: 2 Tangents Tangent & a Secant 1 1 1 y° y° y° x° x° x°

  5. Lengths of Secants, Tangents, & Chords 2 Chords Tangent & Secant 2 Secants

  6. 12-5 Circles in the Coordinate Plane Bonus: Completing the Square

  7. Objectives Write equations and graph circles in the coordinate plane. Use the equation and graph of a circle to solve problems.

  8. The equation of a circle is based on the Distance Formula and the fact that all points on a circle are equidistant from the center.

  9. Example 1A: Writing the Equation of a Circle Write the equation of each circle. J with center J (2, 2) and radius 4 (x – h)2 + (y – k)2 = r2 Equation of a circle Substitute 2 for h, 2 for k, and 4 for r. (x – 2)2 + (y – 2)2 = 42 (x – 2)2 + (y – 2)2 = 16 Simplify.

  10. Example 1B: Writing the Equation of a Circle Write the equation of each circle. K that passes through J(6, 4) and has center K(1, –8) Distance formula. Simplify. Substitute 1 for h, –8 for k, and 13 for r. (x – 1)2 + (y – (–8))2 = 132 (x – 1)2 + (y + 8)2 = 169 Simplify.

  11. Check It Out! Example 1a Write the equation of each circle. P with center P(0, –3) and radius 8 (x – h)2 + (y – k)2 = r2 Equation of a circle Substitute 0 for h, –3 for k, and 8 for r. (x – 0)2 + (y – (–3))2 = 82 x2 + (y + 3)2 = 64 Simplify.

  12. Check It Out! Example 1b Write the equation of each circle. Q that passes through (2, 3) and has center Q(2, –1) Distance formula. Simplify. Substitute 2 for h, –1 for k, and 4 for r. (x – 2)2 + (y – (–1))2 = 42 (x – 2)2 + (y + 1)2 = 16 Simplify.

  13. If you are given the equation of a circle, you can graph the circle by making a table or by identifying its center and radius.

  14. Since the radius is , or 4, use ±4 and use the values between for x-values. Example 2A: Graphing a Circle Graph x2 + y2 = 16. Step 1 Make a table of values. Step 2 Plot the points and connect them to form a circle.

  15. (3, –4) Example 2B: Graphing a Circle Graph (x – 3)2 + (y + 4)2 = 9. The equation of the given circle can be written as (x – 3)2 + (y –(–4))2 = 32. So h = 3, k = –4, and r = 3. The center is (3, –4) and the radius is 3. Plot the point (3, –4). Then graph a circle having this center and radius 3.

  16. Since the radius is , or 3, use ±3 and use the values between for x-values. Check It Out! Example 2a Graph x² + y² = 9. Step 2 Plot the points and connect them to form a circle.

  17. (3, –2) Check It Out! Example 2b Graph (x – 3)2 + (y + 2)2 = 4. The equation of the given circle can be written as (x – 3)2 + (y –(–2))2 = 22. So h = 3, k = –2, and r = 2. The center is (3, –2) and the radius is 2. Plot the point (3, –2). Then graph a circle having this center and radius 2.

  18. Lesson Quiz: Part I Write the equation of each circle. 1.L with center L (–5, –6) and radius 9 (x + 5)2 + (y + 6)2 = 81 2. D that passes through (–2, –1) and has center D(2, –4) (x – 2)2 + (y + 4)2 = 25

  19. Lesson Quiz: Part II Graph each equation. 3.x2+ y2= 4 4. (x – 2)2 + (y + 4)2= 16

  20. Standard Equation of a Circle Review 9.3 Circles center: radius:

  21. Review -coordinate of a point on a circle -coordinate of a point on a circle Radius -coordinate of the center of the circle -coordinate of the center of the circle

  22. Completing the Square Recall that we can solve quadratic equations of the form by “completing the square”… 1. Subtract c from both sides. 2. Square half the coefficient of x and add it to both sides. 3. Then “un-FOIL” We can use this process to find the standard equation of circles when given the “FOILed” equation.

  23. Completing the Square to Find the Equation of a Circle Find the center and radius of the circle .

  24.  Find the center and radius of a circle

  25. FINDING THE CENTER AND RADIUS BY COMPLETING THE SQUARE Example 3 Show that x2 – 6x + y2 +10y + 25 = 0 has a circle as a graph. Find the center and radius. Solution We complete the square twice, once for x and once for y. and

  26. FINDING THE CENTER AND RADIUS BY COMPLETING THE SQUARE Example 3 Add 9 and 25 on the left to complete the two squares, and to compensate, add 9 and 25 on the right. Complete the square. Factor Add 9 and 25 on both sides. Since 9 > 0, the equation represents a circle with center at (3, –5) and radius 3.

  27. FINDING THE CENTER AND RADIUS BY COMPLETING THE SQUARE Example 4 Show that 2x2 + 2y2 – 6x +10y = 1 has a circle as a graph. Find the center and radius. Solution To complete the square, the coefficients of the x2- and y2-terms must be 1. Group the terms; factor out 2.

  28. FINDING THE CENTER AND RADIUS BY COMPLETING THE SQUARE Example 4 Group the terms; factor out 2. Be careful here.

  29. FINDING THE CENTER AND RADIUS BY COMPLETING THE SQUARE Example 4 Factor; simplify on the right. Divide both sides by 2.

  30. FINDING THE CENTER AND RADIUS BY COMPLETING THE SQUARE Example 4 Divide both sides by 2.

  31. Deriving The Quadratic Formula Divide both sides by a Complete the square by adding (b/2a)2 to both sides Factor (left) and find LCD (right) Combine fractions and take the square root of both sides Subtract b/2a and simplify

  32. le fin КОНЕЦ τέλοσ final The end finito ﭙﺎﻴﺎﻥ sof

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