To name coordinates of special figures by using their properties

Transcription

1 6-8 Appling Coordinate Geometr Content tandard Prepares for G.GP.4 Use coordinates to prove simple geometric theorems algebraicall. bjective o name coordinates of special figures b using their properties Knowing previousl established properties of parallelograms will help with this one. he points shown are three vertices of a parallelogram. hat are all the possible coordinates of D, the fourth verte? How do ou know? 6 B 4 C 2 A AHAICAL PACIC In the olve It, ou found coordinates of a point and named it using numbers for the - and -coordinates. In this lesson, ou will learn to use variables for the coordinates. ssential Understanding You can use variables to name the coordinates of a figure. his allows ou to show that relationships are true for a general case. Lesson Vocabular coordinate c proof In Chapter 5, ou learned about the segment joining the midpoints of two sides of a triangle. Here are three possible was to place a triangle and its midsegment. Q(c, d) P(a, b) (e, b) Q(a, b) P(0, 0) (c, 0) Q(0, 2b) P( 2a, 0) (2c, 0) Figure 1 Figure 2 Figure 3 Figure 1 does not use the aes, so it requires more variables. Figures 2 and 3 have good placement. In Figure 2, the midpoint coordinates are a 2, b c 2 and a 2, b 2. In Figure 3, the coordinates are ( a, b) and (c, b). You can see that Figure 3 is the easiest to work with. 406 Chapter 6 Polgons and Quadrilaterals

2 o summarize, to place a figure in the coordinate plane, it is usuall helpful to place at least one side on an ais or to center the figure at the origin. For the coordinates, tr to anticipate what ou will need to do in the problem. hen multipl the coordinates b the appropriate number to make our work easier. How do ou start the problem? Look at the position of the figure. Use the given information to determine how far each verte is from the - and -aes. Problem 1 aming Coordinates hat are the coordinates of the vertices of each figure? A Q is a square where Q 2a. B I is an isosceles triangle where I 2a. he aes bisect each side. he -ais is a median. Q I ince Q is a square centered at he -ais is a median, so it bisects I. the origin and Q 2a, and Q are I 2a, so and I are both a units from the each a units from each ais. he -ais. he height of I does not depend same is true for the other vertices. on a, so use a different variable for. ( a, a) Q(a, a) (0, b) ( a, a) (a, a) ( a, 0) I(a, 0) Got It? 1. hat are the coordinates of the vertices of each figure? a. C is a rectangle with height a b. KI is a kite where I 2a, K b, and length 2b. he -ais bisects and c. he -ais bisects I. C and. I C K Lesson 6-8 Appling Coordinate Geometr 407

3 Problem 2 Using Variable Coordinates he diagram shows a general parallelogram with a verte at the origin and one side along the -ais. hat are the coordinates of D, the point of intersection of the diagonals of ABC? How do ou know? C(2b, 2c) D B(2a 2b, 2c) ABC B bisects AC and AC bisects B A(2a, 0) he coordinates of D ince the diagonals of a parallelogram bisect each other, the midpoint of each segment is their point of intersection. Use the idpoint Formula to find the midpoint of one diagonal. Use the idpoint Formula to find the midpoint of AC. 2a 2b D midpoint of AC 2, 0 2c 2 (a b, c) he coordinates of the point of intersection of the diagonals of ABC are (a b, c). Got It? 2. a. easoning In Problem 2, eplain wh the -coordinate of B is the sum of 2a and 2b. b. he diagram below shows a trapezoid with the base centered at the origin. Is the trapezoid isosceles? plain. ( b, c) A(b, c) ( a, 0) P(a, 0) You can use coordinate geometr and algebra to prove theorems in geometr. his kind of proof is called a coordinate proof. ometimes it is easier to show that a theorem is true b using a coordinate proof rather than a standard deductive proof. It is useful to write a plan for a coordinate proof. Problem 3 shows ou how. 408 Chapter 6 Polgons and Quadrilaterals

4 How do ou start? tart b drawing a diagram. hink about how ou want to place the figure in the coordinate plane. Problem 3 Planning a Coordinate Proof Plan a coordinate proof of the rapezoid idsegment heorem (heorem 6-21). (1) he midsegment of a trapezoid is parallel to the bases. (2) he length of the midsegment of a trapezoid is half the sum of the lengths of the bases. tep 1 Draw and label a figure. tep 2 rite the Given and Prove statements. idpoints will be involved, so use multiples of 2 to name coordinates. (2b, 2c) A(2d, 2c) Use the information on the diagram to write the statements. Given: is the midsegment of trapezoid AP. P(2a, 0) Prove: P, A, 1 2 (P A) tep 3 Determine the formulas ou will need. hen write the plan. and., P, and A are equal. If the are,, P, and A are parallel., P, and A. Got It? 3. Plan a coordinate proof of the riangle idsegment heorem (heorem 5-1). Lesson Check Do ou know H? Use the diagram at the right. 1. In KL, 2a. hat are the coordinates of K and? 2. hat are the slopes of the diagonals of KL? 3. hat are the coordinates of the point of intersection of K and L? K L(2a 2b, c) Do ou UDAD? AHAICAL PACIC 4. easoning How do variable coordinates generalize figures in the coordinate plane? 5. easoning A verte of a quadrilateral has coordinates (a, b). he -coordinates of the other three vertices are a or a, and the -coordinates are b or b. hat kind of quadrilateral is the figure? 6. rror Analsis A classmate sas the endpoints of the midsegment of the trapezoid in Problem 3 are b 2, c a 2 and d 2, c 2. hat is our classmate s error? plain. Lesson 6-8 Appling Coordinate Geometr 409

5 Practice and Problem-olving ercises AHAICAL PACIC A Practice Algebra hat are the coordinates of the vertices of each figure? ee Problem rectangle with base b 8. square with sides of 9. square centered at the origin, and height h length a with side length b 10. parallelogram where is 11. rhombus centered at the 12. isosceles trapezoid with base a units from the origin and origin, with 2r and centered at the origin, with is b units from the origin 2t base 2a and c 13. he diagram below shows a parallelogram. ithout using the Distance Formula, determine whether the parallelogram is a rhombus. How do ou know? ee Problem 2. A( a, a) B(b, b) D( b, b) C(a, a) B Appl 14. Plan a coordinate proof to show that the midpoints of the sides of an isosceles trapezoid form a rhombus. a. ame the coordinates of isosceles trapezoid AP at the right, with bottom base length 4a, top base length 4b, and G 2c. he -ais bisects the bases. D b. rite the Given and Prove statements. c. How will ou find the coordinates of the midpoints of each side? d. How will ou determine whether DFG is a rhombus? 15. pen-nded Place a general quadrilateral in the coordinate plane. 16. easoning A rectangle LP is centered at the origin with (r, s). hat are the coordinates of P? ee Problem 3. G P A F 410 Chapter 6 Polgons and Quadrilaterals

6 Give the coordinates for point P without using an new variables. 17. isosceles trapezoid 18. trapezoid with a right 19. kite (a, b) (a, b) P (0, c) P (c, 0) P (0, a) (b, 0) (0, c) 20. a. Draw a square whose diagonals of length 2b lie on the - and -aes. b. Give the coordinates of the vertices of the square. c. Compute the length of a side of the square. d. Find the slopes of two adjacent sides of the square. e. riting Do the slopes show that the sides are perpendicular? plain. 21. ake two drawings of an isosceles triangle with base length 2b and height 2c. a. In one drawing, place the base on the -ais with a verte at the origin. b. In the second, place the base on the -ais with its midpoint at the origin. c. Find the lengths of the legs of the triangle as placed in part (a). d. Find the lengths of the legs of the triangle as placed in part (b). e. How do the results of parts (c) and (d) compare? 22. and are the midpoints of and, respectivel. In parts (a) (c), find the coordinates of and. a. b. c. (a, b) (c, d) (2a, 2b) (2c, 2d) (4a, 4b) (4c, 4d) (?,?) (e, 0) (?,?) (2e, 0) (?,?) (4e, 0) d. You are to plan a coordinate proof involving the midpoint of. hich of the figures (a) (c) would ou prefer to use? plain. Plan the coordinate proof of each statement. 23. hink About a Plan he opposite sides of a parallelogram are congruent (heorem 6-3). How will ou place the parallelogram in a coordinate plane? hat formulas will ou need to use? 24. he diagonals of a rectangle bisect each other. 25. he consecutive sides of a square are perpendicular. Classif each quadrilateral as precisel as possible. 26. A(b, 2c), B(4b, 3c), C(5b, c), D(2b, 0) 27. (0, 0), P(t, 2s), Q(3t, 2s), (4t, 0) 28. (a, b), F(2a, 2b), G(3a, b), H(2a, b) 29. (0, 0), L( e, f), (f e, f e), (f, e) Lesson 6-8 Appling Coordinate Geometr 411

7 30. hat propert of a rhombus makes it convenient to place its diagonals on the - and -aes? 31. arine Archaeolog arine archaeologists sometimes use a coordinate sstem on the ocean floor. he record the coordinates of points where artifacts are found. Assume that each diver searches a square area and can go no farther than b units from the starting point. Draw a model for the region one diver can search. Assign coordinates to the vertices without using an new variables. C Challenge Here are coordinates for eight points in the coordinate plane (q p 0). A(0, 0), B(p, 0), C(q, 0), D(p q, 0), (0, q), F(p, q), G(q, q), H(p q, q). hich four points, if an, are the vertices for each tpe of figure? 32. parallelogram 33. rhombus 34. rectangle 35. square 36. trapezoid 37. isosceles trapezoid tandardized est Prep A/AC 38. hich number of right angles is possible for a quadrilateral to have? eactl one eactl two eactl three eactl four 39. he vertices of a rhombus are located at (a, 0), (0, b), ( a, 0), and (0, b), where a 0 and b 0. hat is the midpoint of the side that is in Quadrant II? a 2, b 2 a 2, b 2 a 2, b 2 a 2, b In PQ, PQ 35 cm and Q 12 cm. hat is the perimeter of PQ? 23 cm 47 cm 94 cm 420 cm hort esponse 41. In PQ, PQ P Q. ne angle measures 170. List all possible whole number values for m P. ied eview 42. Let X( 2, 3), Y(5, 5), and (4, 10). Is XY a right triangle? plain. rite (a) the inverse and (b) the contrapositive of each statement. ee Lesson 6-7. ee Lesson If 51, then If a 5, then a If b 4, then b is negative. 46. If c 0, then c is positive. 47. If the sum of the interior angle measures of a polgon is not 360, then the polgon is not a quadrilateral. Get ead! o prepare for Lesson 6-9, do ercises 48 and Find the equation for the line that contains the origin and (4, 5). ee Lesson Find the equation for the line that contains (p, q) and has slope a b. 412 Chapter 6 Polgons and Quadrilaterals