To classify polygons in the coordinate plane

Transcription

1 6-7 Polgons in the oordinate Plane ontent Standard G.GP.7 Use coordinates to compute perimeters of polgons... bjective o classif polgons in the coordinate plane ppl what ou learned - about classifing polgons. MHMIL PRIS You and a friend are plaing a board game. Plaers place rubber bands on their own square grid to form different shapes. he object of the game is to guess the vertices of our opponent s shape. How would ou place pieces on the grid shown to complete a right isosceles triangle? Sketch the triangle and justif the placement of each piece. In the Solve It, ou formed a polgon on a grid. In this lesson, ou will classif polgons in the coordinate plane. ssential Understanding You can classif figures in the coordinate plane using the formulas for slope, distance, and midpoint. he chart below reviews these formulas and tells when to use them. Ke oncept Formulas and the oordinate Plane Formula Distance Formula d ( 1 ) ( 1 ) Midpoint Formula M 1, 1 Slope Formula m 1 1 When to Use It o determine whether o determine o determine whether 00 hapter 6 Polgons and Quadrilaterals

2 Problem 1 lassifing a riangle What formulas do ou use? he terms scalene, isosceles, and equilateral have to do with the side lengths of a triangle. Use the Distance Formula. Is scalene, isosceles, or equilateral? he vertices of the triangle are (0, 1), (, ), and (7, 0). Find the lengths of the sides using the Distance Formula. ( 0) ( 1) 16 9 Simplif within parentheses. hen simplif the powers. 6 5 Simplif the radicand. 5 Simplif. (7 ) (0 ) (0 7) (1 0) Since 5, is isosceles. Got It? 1. DF has vertices D(0, 0), (1, ), and F(5, ). Is DF scalene, isosceles, or equilateral? How can ou determine whether D is a rhombus? You can find the length of each side using the Distance Formula, or ou can determine whether the diagonals are perpendicular. he second wa requires fewer steps. Problem Is D a rhombus? plain. lassifing a Parallelogram Step 1 Use the Slope Formula to find the slopes of the diagonals. (0, ) slope of 5 0 ( ) 5 6 slope of D D(, 1) (, 0) Step Find the product of the slopes Since the product of the slopes is not 1, the diagonals are not perpendicular. So D is not a rhombus. 6 (, 5) Got It?. MNPQ has vertices M(0, 1), N( 1, ), P(, 5), and Q(3, ). a. Is MNPQ a rectangle? plain. b. Is MNPQ a square? plain. c. Reasoning Is the triangle in Problem 1 a right triangle? plain. Lesson 6-7 Polgons in the oordinate Plane 01

3 Problem 3 lassifing a Quadrilateral kite is shown at the right. What is the most precise classification of the quadrilateral formed b connecting the midpoints of the sides of the kite? K(, 0), I(0, ), (8, 0), and (0, ) K I he midpoints i of fthe sides of the kite Use the Midpoint Formula to find the coordinates of the vertices of the inner quadrilateral. Draw a diagram to see it. hen classif the figure. Step 1 Find the midpoint of each side of the kite. 0 midpoint of KI, 0 (, ) midpoint of I 0 8, 0 (, ) midpoint of ( ), (, ) 0 ( ) 0 D midpoint of K, (, ) Step Draw a diagram of D. I K D Step 3 lassif D. ( ) 6 Use the definition of distance D 6 on a number line. D ( ) Since opposite sides are congruent, D is a parallelogram. Since and D are both horizontal, and and D are both vertical, the segments form right angles. So, D is a rectangle. Got It? 3. n isosceles trapezoid has vertices (0, 0), (, ), (6, ), and D(8, 0). What special quadrilateral is formed b connecting the midpoints of the sides of D? 0 hapter 6 Polgons and Quadrilaterals

4 Lesson heck Do ou know HW? 1. RI has vertices ( 3, ), R(3, ), and I(0, 0). Is RI scalene, isosceles, or equilateral?. Is QRS below a rectangle? plain. Do ou UNDRSND? 3. Writing Describe how ou would determine whether the lengths of the medians from base angles D and F are congruent. MHMIL PRIS Q R S. rror nalsis student sas that the quadrilateral with vertices D(1, ), (0, 7), F(5, 6), and G(7, 0) is a rhombus because its diagonals D are perpendicular. What is the student s error? F Practice and Problem-Solving ercises MHMIL PRIS Practice Determine whether is scalene, isosceles, or equilateral. plain. See Problem Determine whether the parallelogram is a rhombus, rectangle, square, or none. plain. See Problem. 8. P( 1, ), (0, 0), S(, 0), (3, ) 9. L(1, ), M(3, 3), N(5, ), P(3, 1) 10. R(, 3), S(, 0), (3, ), V( 3, 1) 11. G(0, 0), H(6, 0), I(9, 1), J(3, 1) 1. W( 3, 0), I(0, 3), N(3, 0), D(0, 3) 13. S(1, 3), P(, ), (3, 1), (0, 0) What is the most precise classification of the quadrilateral formed b connecting in order the midpoints of each figure below? See Problem parallelogram PR 15. rectangle FGH 16. isosceles trapezoid JKLM P R F G H M J K L Lesson 6-7 Polgons in the oordinate Plane 03

5 ppl Graph and label each triangle with the given vertices. Determine whether each triangle is scalene, isosceles, or equilateral. hen tell whether each triangle is a right triangle. 17. (1, 1), R(3, 8), I(6, ) 18. J( 5, 0), K(5, 8), L(, 1) 19. (3, ), ( 10, ), ( 5, 8) 0. H(1, ), ( 1, ), F(5, 6) Graph and label each quadrilateral with the given vertices. hen determine the most precise name for each quadrilateral. 1. P( 5, 0), Q( 3, ), R(3, ), S(5, 0). S(0, 0), (, 0), U(3, ), V( 1, ) 3. F(0, 0), G(5, 5), H(8, ), I(7, 1). M( 1, ), N(1, 6), P(3, 9), Q( 1, 11) 5. (3, 5), (7, 6), (6, ), D(, 1) 6. N( 6, ), P( 3, 1), Q(0, ), R( 3, 5) 7. J(, 1), K(5, ), L(8, 1), M(, 3) 8. H(, 3), I(, 0), J(3, ), K( 3, 1) 9. W( 1, 1), X(0, ), Y(1, 1), Z(0, ) 30. D( 3, 1), ( 7, 3), F(6, 3), G(, 1) 31. hink bout a Plan re the triangles at the right congruent? How do ou know? Which triangle congruence theorem can ou use? Which formula should ou use? 3. Reasoning quadrilateral has opposite sides with equal slopes and consecutive sides with slopes that are negative reciprocals. What is the most precise classification of the quadrilateral? plain. Q P R 6 W S Determine the most precise name for each quadrilateral. hen find its area. 33. (0, ), (, ), ( 3, ), D( 7, ) 3. J(1, 3), K(3, 1), L(7, 1), M(5, 5) 35. D is a midsegment of at the right. Show that the riangle Midsegment heorem holds true for. 36. a. Writing Describe two was ou can show whether a quadrilateral in the coordinate plane is a square. b. Reasoning Which method is more efficient? plain. D Interior Design Interior designers often use grids to plan the placement of furniture in a room. he design at the right shows four chairs around a coffee table. he designer plans for cutouts of chairs on lattice points. She wants the chairs oriented at the vertices of a parallelogram. Does she need to fi her plan? If so, describe the change(s) she should make. 0 hapter 6 Polgons and Quadrilaterals

6 hallenge 38. Use the diagram at the right. a. What is the most precise classification of D? b. What is the most precise classification of FGH? c. re D and FGH congruent? plain. 39. oordinate Geometr he diagonals of quadrilateral FGH intersect at D( 1, ). FGH has vertices at (, 7) and F( 3, 5). What must be the coordinates of G and H to ensure that FGH is a parallelogram? D 6 H F G 6 he endpoints of are ( 3, 5) and (9, 15). Find the coordinates of the points that divide into the given number of congruent segments n Standardized est Prep S/ 5. K( 3, 0), I(0, ), and (3, 0) are three vertices of a kite. Which point could be the fourth verte? (0, 5) (0, 0) (0, ) (0, 10) 6. In the diagram, lines and m are parallel. What is the value of? ( 11) 155 m Short Response 7. In the diagram, which segment is shortest? PS PR PQ QR 8. ( 3, 1), ( 1, ), and (, 1) are three vertices of quadrilateral D. ould D be a rectangle? plain. S 6 P R 57 Q Mied Review 9. lgebra Find the measure of each angle and the value of in the isosceles trapezoid. Find the circumcenter of (1, 1), (5, 3), (5, 1) 51. ( 5, 0), ( 1, 8), ( 1, 0) See Lesson 6-6. See Lesson 5-3. Get Read! o prepare for Lesson 6-8, do ercises 5 5. Find the slope of XY. 5. X(0, a), Y( a, a) 53. X( a, b), Y(a, b) See Lesson X(a, 0), Y(c d, b) Lesson 6-7 Polgons in the oordinate Plane 05