Applications. 12 The Shapes of Algebra. 1. a. Write an equation that relates the coordinates x and y for points on the circle.

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1 Applications 1. a. Write an equation that relates the coordinates and for points on the circle. 1 8 (, ) 1 8 O (13, 0) b. Find the missing coordinates for each of these points on the circle. If there is more than one possible point, give the missing coordinate for each possibilit. Show that each ordered pair satisfies the equation. (0, j) (5, j) (-, j) (-8, j) (j, 10) (j, -6) (j,0) (j, -) c. Write an inequalit that relates the coordinates and for points inside the circle. d. Choose an point in the interior of the circle and confirm that this point is a solution for the inequalit ou wrote in part (c). e. Choose an point outside the circle and check that it is not a solution for the inequalit ou wrote in part (c). 1 The Shapes of Algebra

2 . a. Write an equation that relates the coordinates and for points on the circle. 1 8 O 1 8 (10, 0) (, ) 8 1 b. Find the missing coordinates for each of these points on the circle. If there is more than one possible point, give the missing coordinate for each possibilit. Show that each ordered pair satisfies the equation. (8, j) (3, j) (-, j) (0, j) (j, -) (j, -6) (j,0) (j,) c. Write an inequalit that describes the points in the interior of the circle. d. Write an inequalit that describes the points outside the circle. e. Choose one point in the interior of the circle and one point outside the circle and confirm these are solutions for the appropriate inequalities. For: Algebra Tools Visit: PHSchool.com Web Code: apd-710 Investigation 1 Equations for Circles and Polgons 13

3 3. On a cop of this diagram, draw quadrilaterals meeting the conditions in parts (a) (d). Your figures should fit entirel on the grid and should not overlap. a. Rectangle ABCD lies entirel in the second quadrant. b. Rectangle EFGH lies entirel in the A first quadrant. c. Rectangle PQRS is not a square. It lies entirel in the third quadrant ecept for verte Q. d. Square TUVW lies entirel in the fourth quadrant. B 8 6 For: Help with Eercise 3 Web Code: ape-7103 Q 8 6 O 6 8 P E F 6 U 8 T. The quadrilaterals named in parts (a) (d) are parallelograms formed on the diagram at the right. Give the coordinates for the fourth verte. Then, calculate the slopes of the sides to show that the opposite sides are parallel. a. JKLM b. NPQR c. STUV d. WYXZ J P Q 8 6 K L N U 8 6 O 6 8 T Y X 6 S 8 Z 1 The Shapes of Algebra

4 Find the equation of a line parallel to the given line. 5. = = = = = = For Eercises 11 16, find the equation of a line perpendicular to the given line = = = = = = a. The circle in this design is centered at the origin. Find coordinates for points J, K, and L. S(0, 6) P R L J T V K b. Points P, R, V, and T are the midpoints of the segments on which the lie. Find coordinates for each of these points. c. Find coordinates of the vertices of the innermost quadrilateral. Is this quadrilateral a square? Eplain. Find the midpoint of the segment with the given endpoints. 18. (0, 0) and (, 6) 19. (3, ) and (7, -) 0. (1, ) and (8, 5) 1. (1, ) and (-5, 6). (0, 0) and (-, -7) 3. (-1, -5) and (-6, ) Investigation 1 Equations for Circles and Polgons 15

5 Connections Use the Pthagorean Theorem to find the unknown side length.. 5. For: Multiple-Choice Skills Practice Web Code: apa cm 9.6 cm 1. cm 1 cm cm 13.5 cm.3 cm 9. cm Write an equation for the line with the given slope and -intercept slope -intercept (0, 3), 1 9. slope - -intercept (0, 5) 3, slope 6, -intercept (0, ) Write an equation for the line with the given slope and that passes through the given point. 31. slope, point (3, 1) 3. slope -, point (-1, 7) slope - point (0, 5) 6, 16 The Shapes of Algebra

6 3. For each tpe of quadrilateral in the first column, identif all the properties from the second column that appl to that tpe of quadrilateral. Quadrilateral Tpes Properties a. square i. Two pairs of parallel sides b. rectangle ii. Four right angles c. rhombus iii. Two pairs of congruent sides d. parallelogram iv. Interior angle measures with a sum of 360 o v. Opposite angle measures with a sum of 180 o vi. Perpendicular diagonals For Eercises 35 6, find the value of each epression (-18) (-19) (-1) (-) (-3). 5 3 (-13) (-) 6. 8 (-) 7. Suppose ou ve drawn a design on a coordinate grid. Tell whether each coordinate rule will produce a similar design. a. (, ) S ( +, + 3) b. (, ) S (,3) c. (, ) S (.5,.5) d. (, ) S (-, -) 8. The radius of this crop design is 6 meters. a. What is the area of the smaller square? b. What is the area of the region between the smaller and larger squares? c. What is the area of the region between the larger square and the circle? d. Describe all the smmetries in the design. 9. a. Consider the points A(-, ), B(-1, -1), C(-1, ), D(0, -3), E(0, ), F(1, 0), G(, 0), H(, -1), J(5, -1), K(6, -1.5). Without plotting points or drawing lines, find the slope of these lines. line AB line CD line EF line GH line JK b. Order the slopes in part (a) from least to greatest. Investigation 1 Equations for Circles and Polgons 17

7 50. a. Suppose ou connect the midpoints of the sides of a triangle as shown below to form a smaller triangle. How does the perimeter of the blue triangle compare to that of the original triangle? b. How does the area of the blue triangle compare to that of the original triangle? 51. Two students became intrigued b crop designs. The did a project comparing the occurrences of different shapes in three countries, A, B and C. CROP CIRCLE OCCURRENCES Boundar Tpe Countr A Countr B Countr C Circle Square a. Make a circle graph to compare the total number of circular crop designs in three countries with the total number of square crop designs. b. Make a bar graph to compare the crop designs from countries A, B, and C. c. Make three statements summarizing the students findings on crop designs in the three countries. Find the equation of the line through the points. 5. (, 3) and (0, 1) 53. (-1, 3) and (, -9) 5. (-1, -1) and (3, 7) 18 The Shapes of Algebra

8 55. Kara started to find the midpoints of some segments, but she didn t finish. Her work is shown in parts (a) (c). Finish her calculations to find the midpoint. Then give the coordinates of the segment s endpoints. a. Q 3 1 9, R b. Q 3, R c. Q 3 1 (9), 1 1 (1) R For Eercises 56 58, tell whether the lines intersect. If the do, find their intersection point both algebraicall and graphicall. If the don t intersect, eplain how ou know. 56. = - 11 and = = + 10 and = = and = 3( + 10) 59. Multiple Choice Which epression is equivalent to ? A. 3( + 10) B C. 5( + ) - D Etensions 60. This circle has radius 5 and center (1, ). Find or estimate the missing coordinates for these points on the circle. In each case, use the Pthagorean Theorem to check that the point is 5 units from the center. a. (j,6) b. (5, j) 6 c. (-3, j) d. (1, j) e. (j,) f. (, j) (1, ) O 6 Investigation 1 Equations for Circles and Polgons 19

9 61. a. This circle has radius 5 and center (1, ). AC is parallel to the -ais. BC is parallel to the -ais. What are the lengths of AC, BC, AB in terms of and? b. What equation shows how these side A(1, ) lengths are related? O c. Suppose ou redraw the figure with B(, ) in a different position, but still on the circle. Would the coordinates of B still fit the equation ou wrote in part (b)? d. Based on this eample, what do ou think is the general equation for points on a circle with center (m, n) and radius r? C B(, ) 6. a. The vertices of the blue triangle are the midpoints of the sides of #FGH. How are the sides of the blue triangle related to those of #FGH? Use coordinates to check our ideas. b. Draw several more triangles and connect their midpoints to form a smaller triangle. Record our observations. G H F 63. Consider the points O(0, 0), X(, 5), L(, 3), and M(6, 8). a. Points U and V divide OX into three equal-length segments. Find the coordinates of points U and V. b. Points W and Z divide LM into three equal-length segments. Find the coordinates of points W and Z. c. OX can be translated to correspond with LM. Describe the rule for this translation. d. Check our coordinates for points W and Z b appling our translation rule to points U and V. 0 The Shapes of Algebra

10 6. Use the diagram below. Record our answers to parts (a) (c) in a cop of the table at the bottom of the page C 6 B A a. Find the coordinates of points X and Y that divide AC into three equal-length segments. b. Find the coordinates of points M and N that divide BC into three equal-length segments. c. Find the coordinates of points P and Q that divide AB into three equal-length segments. d. Describe the pattern relating the coordinates of the endpoints to the coordinates of the two points that divide the segment into thirds. e. How can ou find the coordinates of the two points R and S that divide the segment joining points G( 1, 1 ) and H(, ) into three equal-length segments? Segment Endpoint Dividing Point Dividing Point Endpoint AC A(, ) X(, ) Y(, ) C(, ) BC B(, ) M(, ) N(, ) C(, ) AB A(, ) P(, ) Q(, ) B(, ) Investigation 1 Equations for Circles and Polgons 1

11 65. Multiple Choice In triangle ABC, point D is on AB and point E is on AC, such that AD = DB and AE = EC. AD means twice the length of AD and AE means twice the length of AE. Which of the following statements is not true? A. #ADE is similar to #ABC B. BC = 3DE C. DE is parallel to BC D. area of #ABC = 3(area of #ADE) 66. In this diagram, the vertices of PQRS are the midpoints of the sides of quadrilateral WXYZ. WY is twice as long as SR. Z S R Y W P X Q a. Eplain wh #WZY is similar to #SZR. b. How does the similarit of #WZY and #SZR impl that SR is parallel to WY? c. How could ou show that PQ is parallel to WY? d. Wh do the results of parts (b) and (c) impl that SR is parallel to PQ? e. How could ou repeat the reasoning from parts (a) (d) to show that SP is parallel to RQ? f. How does the reasoning from parts (a) (e) show that PQRS is a parallelogram? The Shapes of Algebra