Unit 5 Lesson 1 Investigation PQRS 3 =

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Name: Check Your Understanding Unit 5 Lesson 1 Investigation PQRS 3 = Consider quadrilateral PQRS with verte matri 8 28 24 4 4 12 28 20. a. Draw quadrilateral PQRS on a coordinate grid. b.

1 Name: Check Your Understanding Unit 5 Lesson 1 Investigation PQRS 3 = Consider quadrilateral PQRS with verte matri a. Draw quadrilateral PQRS on a coordinate grid. b. What special kind of quadrilateral is PQRS? Use coordinates to justif our answer. Investigation 3 Representing and Reasoning with Circles In Investigations 1 and 2, ou learned how to represent and analze polgons in a coordinate plane. You can describe their sides using linear equations and stud their properties using ideas of distance and slope. Polgons, particularl triangles and quadrilaterals, are the building blocks for architectural designs. Industrial, automotive, and aerospace designs often require that shapes have circular components. Your work on the problems in this investigation will help ou answer these questions: What information is needed to create a circle in a coordinate plane? How can ou represent circles in a coordinate plane with equations? How can ou use general coordinates of points to reason about special properties of circles? LESSN 1 A Coordinate Model of a Plane 175

CPMP-Tools 1 As a class, eplore how interactive geometr software could be used to create the design shown at the right. a. What information was needed b the software to draw each circle?

2 CPMP-Tools 1 As a class, eplore how interactive geometr software could be used to create the design shown at the right. a. What information was needed b the software to draw each circle? Wh do ou think that information is sufficient? b. Clear the window and redraw the square, centered at the origin, with side length 10 units. c. Draw a circle inscribed in the square, that is, a circle that touches each side of the square at one point. Describe the points of contact of the circle and square. d. Draw a circle circumscribed about the square, that is, a circle that passes through each verte of the square. e. What is the radius of each circle in Parts c and d? 2 Here are two circles with center at the origin and radius 10 drawn in a coordinate plane. Diagram I D(?,?) C(a, a) B(8,?) Diagram II P(, ) A(?,?) F(?, -5) E(-2,?) a. What must be true about the distance from point to an other point on the circle? b. Without the help of software, find the missing coordinate(s) of points A through F on the circle in Diagram I. c. Suppose P(, ) is an point on the circle in Diagram II. i. What must be true about the distance P? ii. Write an equation showing the relationship between,, and the radius of the circle. d. Write an equation for a circle with its center at the origin and with radius 7. With radius 3. With radius r. 176 UNIT 3 Coordinate Methods

3 A calculator-produced circle is shown below. The Zsquare window has a scale on both aes of 1 unit. a. What is the radius of the circle? b. Write an equation for this circle. c. What epressions could be placed in the Y= menu to produce the circle?

4 Some of the circles ou created in Problem 1 did not have their centers at the origin. However, ou can use reasoning similar to that in Problem 2 to find equations for these circles. a. What is the center and radius of the circle whose center is on the positive -ais?

3 3 A calculator-produced circle is shown below. The Zsquare window has a scale on both aes of 1 unit. a. What is the radius of the circle? b. Write an equation for this circle. c. What epressions could be placed in the Y= menu to produce the circle? Do our epressions produce a circle with the same radius? d. Use our calculator to produce a cop of the circle shown in the computer displa on page Some of the circles ou created in Problem 1 did not have their centers at the origin. However, ou can use reasoning similar to that in Problem 2 to find equations for these circles. a. What is the center and radius of the circle whose center is on the positive -ais? i. Suppose P(, ) is an point on that circle. Eplain wh it must be the case that ( - 5) = 5. ii. Use that information to write an equation for the circle that does not involve a radical smbol. b. Write similar equations for: i. the circle whose center is on the positive -ais. ii. the circle whose center is on the negative -ais. iii. the circle whose center is on the negative -ais. c. Verif that the coordinates of the vertices of the square satisf our equations of the four circles that contain those vertices. Share the workload with our classmates. 5 Now tr to generalize our work in Problems 2 4 to a circle whose center is not on an ais. a. Use reasoning similar to that in P(, ) Problem 4 to find the equation of a circle with center C(h, k) and radius r. r b. Compare our equation with those of C(h, k) our classmates. Resolve an differences. c. Rewrite our equation in Part b for the case when C(h, k) is the origin. What do ou notice? LESSN 1 A Coordinate Model of a Plane 177

4 6 Without using technolog, determine which of the following equations describe a circle in a coordinate plane. For each equation that represents a circle, determine the center, the radius, and one point on the circle. For each equation that does not represent a circle, eplain wh not. a = 25 b. 2 + = 16 c = 108 d. ( - 5) 2 + ( - 1) 2 = 81 e = 9 f. 2 + ( + 5) 2 = 1 CPMP-Tools 7 Coordinates as emploed b interactive geometr software open new windows to geometr b allowing ou to easil create figures and search for patterns in them. Complete Parts a c using our software. You can create the figures ourself or use the Eplore Angles in Circles custom tool. a. Draw a circle with center A and diameter with endpoint B. Label the other endpoint C. b. Construct a new point D on the circle. Then draw BD and CD. c. Click and drag point D along the circumference of the circle. i. What appears to be true about CDB in all cases? ii. How is our conjecture supported b calculations from the Measurements menu? d. State our conjecture in the form: An angle inscribed in a semicircle.... Compare our conjecture with our classmates and resolve an differences. 8 As ou saw in Investigation 2, coordinates can provide a powerful wa to justif conjectures ou make about geometric figures. The ke is to position the figure in a coordinate plane so that general coordinates are eas to work with. A circle with center at the origin and radius r is shown below. Point A(a, b) is a general point on the circle, different from points P and Q which are endpoints of a diameter on the -ais. A(a, b) P(-r, 0) r Q(r, 0) 178 UNIT 3 Coordinate Methods

5 Use these general coordinates and the following questions to help justif the conjecture ou made in Problem 7: An angle inscribed in a semicircle is a right angle. a. What are some possible methods ou could use to justif that PAQ is a right angle? b. What are the coordinates of points P and Q? c. Since point A(a, b) is on the circle, what must be true about the distance A? How is that distance related to the coordinates a and b? d. Stud Jack s argument below. He shows that PAQ is a right triangle, and so PAQ is a right angle. Check the correctness of Jack s reasoning and give reason(s) justifing each step. If there are an errors in Jack s reasoning, correct them. Jack s argument The length of PA = (a + r) 2 + b 2, so (PA) 2 = (a + r) 2 + b 2. (1) The length of AQ = (r a) 2 + b 2, so (AQ) 2 = (r a) 2 + b 2. (2) The length of PQ = 2r, so (PQ) 2 = 4r 2. (3) (PA) 2 + (AQ) 2 = (a + r) 2 + b 2 + (r a) 2 + b 2 (4) = (a 2 + 2ar + r 2 + b 2 ) + (r 2 2ar + a 2 + b 2 ) (5) = 2a 2 + 2r 2 + 2b 2 (6) = 2(a 2 + b 2 ) + 2r 2 (7) = 2r 2 + 2r 2 (8) = 4r 2 (9) = (PQ) 2 (10) Therefore, PAQ is a right triangle with PAQ a right angle. (11) e. Now eamine Malaa s argument justifing the conjecture that PAQ is a right angle. Check the correctness of Malaa s reasoning and give reason(s) justifing each step. Correct an errors in Malaa s reasoning. Malaa s argument The slope of PA is The slope of QA is b_ a + r. (1) b_ a r. ) (2) a r = _ b 2 a 2 r. (3) 2 Since a 2 + b 2 = r 2, it follows that a 2 r 2 = b 2. (4) The product of the slopes is ( b _ a + r ) ( b This means that the product of the slopes is b2 = 1. (5) b2 So, PA AQ and PAQ is a right angle. (6) LESSN 1 A Coordinate Model of a Plane 179

15.4 Equation of a Circle

15.4 Equation of a Circle Name Class Date 1.4 Equation of a Circle Essential Question: How can ou write the equation of a circle if ou know its radius and the coordinates of its center? Eplore G.1.E Show the equation of a circle