The Coordinate Plane. Circles and Polygons on the Coordinate Plane. LESSON 13.1 Skills Practice. Problem Set

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1 LESSON.1 Skills Practice Name Date The Coordinate Plane Circles and Polgons on the Coordinate Plane Problem Set Use the given information to show that each statement is true. Justif our answers b using theorems and b using algebra. 1. The center of circle O is at the origin. The coordinates of the given points are A(, ), B(, ), and C(, ). Show that n ABC is a right triangle. n ABC is an inscribed triangle in circle O with the hpotenuse as the diameter of the circle, therefore the triangle is a right triangle b the Right Triangle Diameter Theorem. AB 5 ( ) 1 ( ) 5 ( ) 5 5 AC 5 ( ) 1 ( ) 5 () 1 () BC 5 ( ) 1 ( ) 5 1 () ( ) 1 ( ) A C O B D Therefore, b the Converse of the Pthagorean Theorem, nabc is a right triangle. Chapter Skills Practice 75

2 LESSON.1 Skills Practice page. The center of circle O is at the origin. AZ and AT are tangent to circle O. The coordinates of the given points are A(, 3), T(, ), and Z(, ). Show that the lengths of AT and AZ are equal. A 3 T 3 Z O The center of circle C is at the origin. AB is tangent to circle C at (3, ). Show that the tangent line is perpendicular to CA. B ( 1, 7) A (3, ) C (, ) 7 Chapter Skills Practice

3 LESSON.1 Skills Practice page 3 Name Date. The center of circle O is at the origin. The coordinates of the given points are A(3, ), B(, 3), D(, 5), E( 3, ), and F( 1.5,.5). Show that EF FD AF FB. E A O F B D Chapter Skills Practice 77

4 LESSON.1 Skills Practice page 5. The center of circle O is at the origin. The coordinates of the given points are A( 5, 5), B(, 3), C(, 5), and D(, 5). Show that AD AB AC. A D B O C 7 Chapter Skills Practice

5 LESSON.1 Skills Practice page 5 Name Date. The center of circle O is at the origin. BD is perpendicular to OA at point C. The coordinates of the given points are A( 5, ), B(, 3), C(, ), and D(, 3). Show that OA bisects BD. B A C O D Chapter Skills Practice 79

6 LESSON.1 Skills Practice page Classif the polgon formed b connecting the midpoints of the sides of each quadrilateral. Show all our work. 7. The rectangle shown has vertices A(, ), B(, ), C(, ), and D(, ). Midpoint AB : U (, ) Midpoint BC : S (, ) Midpoint CD : T (, ) Midpoint DA : R (, ) D (, ) R T C (, ) S A (, ) U B (, ) Slope RT Slope US Slope TS 5 Slope RU RT and US are parallel since the have the same slope of. TS and RU are parallel since the have the same slope of. There are no perpendicular sides. The slopes are not opposite reciprocals. RT 5 ( ) 1 ( ) TS 5 ( ) 1 ( ) 5 ( ) 1 ( ) 5 ( ) 1 ( ) US 5 ( ) 1 ( ) 5 ( ) 1 ( ) 5 1 RU 5 ( ) 1 ( ) 5 ( ) 1 ( ) 5 1 All four sides of RTSU are congruent. Opposite sides are parallel and all sides are congruent, so the quadrilateral formed b connecting the midpoints of the rectangle is a rhombus. Chapter Skills Practice

7 LESSON.1 Skills Practice page 7 Name Date. The isosceles trapezoid shown has vertices A(, ), B(, ), C(, 3), and D(, 3). D (, 3) C (, 3) A (, ) B (, ) Chapter Skills Practice 1

8 LESSON.1 Skills Practice page 9. The parallelogram shown has vertices A(, 5), B(7, 5), C(3, ), and D(, ). A B D C Chapter Skills Practice

9 LESSON.1 Skills Practice page 9 Name Date. The rhombus shown has vertices A(, ), B(, 5), C(, ), and D(, 5). A D B C Chapter Skills Practice 3

10 LESSON.1 Skills Practice page 11. The square shown has vertices A(, ), B(, ), C(, ), and D(, ). D C B A Chapter Skills Practice

11 LESSON.1 Skills Practice page 11 Name Date 1. The kite shown has vertices A(, ), B(, ), C(, 3), and D(, ). A D B C Chapter Skills Practice 5

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13 LESSON. Skills Practice Name Date Bring On the Algebra Deriving the Equation for a Circle Problem Set Write an equation in standard form of each circle. 1. a circle with center point at the origin when r r a circle with center point at the origin when r a circle with center point (, 5) when r 5 1. a circle with center point (, 1) when r 5 7 Chapter Skills Practice 7

14 LESSON. Skills Practice page 5. (, ) (, ). (3, 7) (, 3) Write the equation of each circle in standard form. Then identif the center point and radius of the circle ( 1 1 9) 1 ( 1 1) ( 1 3) 1 ( 1) 5 9 center: (3, 1), radius: 3 Chapter Skills Practice

15 LESSON. Skills Practice page 3 Name Date Chapter Skills Practice 9

16 LESSON. Skills Practice page Determine if each equation represents a circle. If so, describe the location of the center and radius ( 1 1 ) 1 ( 1 1 ) ( 1 ) 1 ( 1 ) 5 5 center: (, ), radius: Chapter Skills Practice

17 LESSON. Skills Practice page 5 Name Date Determine an equation of the circle that meets the given conditions. 17. Same center as circle A, ( 1 3) 1 ( 1 5) 5 9, but with a circumference that is twice that of circle A The radius of circle A is 9, or 3. To calculate the circumference of A, substitute 3 for r in the formula for the circumference of a circle. C 5 r 5 (3) C 5 A circle with twice the circumference of circle A has circumference ( ), or 1 units. To calculate its radius, substitute 1 for C in the formula for the circumference of a circle, and then solve for r. C 5 r 1 5 r 5 r The radius of the circle is. So an equation of the circle with the same center as circle A but with a circumference that is twice that of circle A is ( 1 3) 1 ( 1 5) 5, or ( 1 3) 1 ( 1 5) Same center as circle B, ( ) 1 ( 1 ) 5 9, but with a circumference that is three times that of circle B Chapter Skills Practice 91

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19 LESSON.3 Skills Practice Name Date Is That Point on the Circle? Determining Points on a Circle Problem Set For each circle A, determine whether the given point P lies on the circle. Eplain our reasoning. 1. P (5, 3) r 3 A (, ) 5 a 1 b 5 c r r 3 5 r r 5 3 < 5. Because the length of line segment AP is approimatel 5. units instead of units, line segment AP is not a radius of circle A; therefore, point P does not lie on circle A.. P (, ) A (, ) Chapter Skills Practice 93

20 LESSON.3 Skills Practice page 3. P (5, Œ39) A (, ). A (, ) P (, 7) 9 Chapter Skills Practice

21 LESSON.3 Skills Practice page 3 Name Date P (3, 11 Œ3 ) 5 1 A (1, 1) P (7, ) A (, 3) Chapter Skills Practice 95

22 LESSON.3 Skills Practice page Use smmetr to determine the coordinates of each labeled point on the circle. Give eact values, not approimations P (, ) 1 P (17, 5 Œ3 ) (, ) B (, ) (, ) B (, ) (, ) (3, 15) 15 9 C(, 3) C (3, ) (3 Œ, Œ) (, ) (1, ) (, 7.5) (, ) (1, ) (5, 1) Chapter Skills Practice

23 LESSON. Skills Practice Name Date The Parabola Equation of a Parabola Vocabular 1. locus of points a. A 1 D 5 or B 1 C 5. parabola 3. focus of a parabola. directri of a parabola 5. general form of a parabola. standard form of a parabola 7. ais of smmetr. verte of a parabola 9. concavit b. 5 p or 5 p c. describes the orientation of the curvature of the parabola d. a set of points in a plane that are equidistant from a fied point and a fied line e. the maimum or minimum point of a parabola f. a set of points that share a propert g. a line that passes through the parabola and divides the parabola into two smmetrical parts that are mirror images of each other h. the fied point from which all points of a parabola are equidistant i. the fied line from which all points of a parabola are equidistant Chapter Skills Practice 97

24 LESSON. Skills Practice page Problem Set Determine the equation of the parabola (, ) 1 (, 1) (, 1) ( ) 1 (1 ) 5 ( ) 1 (1 ) 1 (1 ) 5 (1 ) 1 (1 ) 5 (1 ) (, ) 3 (3, ) 5 1 (3, ) Chapter Skills Practice

25 LESSON. Skills Practice page 3 Name Date 3. 5 (, ) 3 (, ) 5 1 (, ) (,.5) (,.5) (, ) 3 5 Chapter Skills Practice 99

26 LESSON. Skills Practice page Identif the verte, ais of smmetr, value of p, focus and directri for each parabola Verte: (, ) Ais of smmetr: 5 Value of p:.75 Focus: (,.75) Directri: (, 7.5) 3 5. (, ) 9 Chapter Skills Practice

27 LESSON. Skills Practice page 5 Name Date 7. Parabola c Chapter Skills Practice 91

28 LESSON. Skills Practice page Sketch each parabola Chapter Skills Practice

29 LESSON. Skills Practice page 7 Name Date Chapter Skills Practice 93

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31 LESSON.5 Skills Practice Name Date Simpl Parabolic More With Parabolas Problem Set Determine the equation of the parabola (, ) (, ) 3 5 (, ) ( ) 1 ( ) 5 ( ) 1 ( ) 1 ( ) 5 ( ) 1 ( ) 5 ( ) (3, 5) (, ) (, 3) Chapter Skills Practice 95

32 LESSON.5 Skills Practice page 3. (5, 1) 1 1 (, ) (, 5). (, ) 1 1 (, ) (3, ) 9 Chapter Skills Practice

33 LESSON.5 Skills Practice page 3 Name Date Identif the verte, ais of smmetr, value of p, focus and directri for each parabola Verte: (3, ) 1 (3, ) Ais of smmetr: 5 3 Value of p: Focus: (3, ) Directri: (, ) Chapter Skills Practice 97

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35 LESSON.5 Skills Practice page 5 Name Date Complete the table for each equation. Then, plot the points and graph the curve on the coordinate plane Chapter Skills Practice 99

36 LESSON.5 Skills Practice page Chapter Skills Practice

37 LESSON.5 Skills Practice page 7 Name Date Rewrite the equations in standard form ( ) 5 3( 1 ) Chapter Skills Practice 911

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