Polygons in the Coordinate Plane. isosceles 2. X 2 4

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1 Name lass ate 6-7 Practice Form G Polgons in the oordinate Plane etermine whether k is scalene, isosceles, or equilateral. 1. isosceles. scalene 3. scalene. isosceles What is the most precise classification of the quadrilateral formed b connecting in order the midpoints of each figure below? 5. parallelogram 6. square 7. rectangle 8. P rhombus S Q H F R G Prentice Hall Gold Geometr Teaching Resources opright b Pearson ducation, Inc., or its affiliates. ll Rights Reserved. 63

2 Name lass ate 6-7 Practice (continued) Form G Polgons in the oordinate Plane 9. Writing escribe two was in which ou can show whether a parallelogram in the coordinate plane is a rectangle. etermine whether the diagonals are congruent using the istance Formula or determine if consecutive sides are perpendicular using the Slope Formula. 10. Writing escribe how ou can show whether a quadrilateral in the coordinate plane is a kite. etermine whether two pairs of consecutive sides are congruent, using the istance Formulas, and that the pairs are not congruent to each other. Use the trapezoid at the right for ercises 11 and Is the trapezoid an isosceles trapezoid? plain. No; no sides are congruent. 1. Is the quadrilateral formed b connecting the midpoints of the trapezoid a parallelogram, rhombus, rectangle, or square? plain. Parallelogram; the slopes of the opposite sides are equal, but adjacent sides are not perpendicular and the sides are not all congruent. etermine the most precise name for each quadrilateral. Then find its area. 13. (6, 3), (, 0), (, 5), (6, ) rhombus; 0 units 3 1. (1, 8), (, 6), (1, ), (, 0) parallelogram; 30 units 15. (3, ), (8, 1), (, 9), (3, 6) rectangle; 68 units 16. (0, 1), (1, ), (, 3), (3, ) parallelogram; 16 units 17. (5, 1), (, 11), (5, 8), (8, 11) square; 18 units etermine whether the triangles are congruent. plain W No; corresponding sides not. F es; corr. sides are. Prentice Hall Gold Geometr Teaching Resources opright b Pearson ducation, Inc., or its affiliates. ll Rights Reserved. 6

3 Name lass ate 6-8 Practice Form G ppling oordinate Geometr lgebra What are the coordinates of the vertices of each figure? 1. rectangle with. rectangle centered at the base b and height h origin with base b and height h (b, h); (0, h); (0, 0); (b, 0) (b, h); F(b, h); G(b, h); (b, h) 3. square with height. parallelogram with height m and point distance j from the origin G F H I W K J H(, ); I(0, ); J(0, 0); K(, 0) Sample: W(0, m); ( j, m); (, 0); (j, 0) 5. kite MNP where PN 5 s 6. isosceles n where 5 n and the -ais bisects PN and the -ais is the median P M N Sample: M(0, b); N(s, 0); (0, c); P(s, 0) Sample: (n, 0); (n, 0); (0, ) 7. How can ou determine if a triangle on a coordinate grid is an isosceles triangle? Use the istance Formula to compare side lengths. 8. How can ou determine if a parallelogram on a coordinate grid is a rhombus? nswers ma var. Sample: Use the Slope Formula to determine if the diagonals are perpendicular. 9. How can ou determine if a parallelogram on a coordinate grid is a rectangle? nswers ma var. Sample: Use the istance Formula to determine if the diagonals are congruent. Prentice Hall Gold Geometr Teaching Resources opright b Pearson ducation, Inc., or its affiliates. ll Rights Reserved. 73

4 Name lass ate 6-8 Practice (continued) Form G ppling oordinate Geometr 10. In the triangle at the right, is at (m 1 r, s), is at (m, p), and is at (r, p). Is this an isosceles triangle? plain. es; 5. oth are equal to "m rm 1 r 1 s 1 sp 1 p. 11. Is the trapezoid shown at the right an isosceles trapezoid? plain. No; the diagonals are not congruent. 5 "(z 1 ) 1 h, F 5 "(z 1 3) 1 h ( z, h) F( z 3, 0) (z, h) (z, 0) For ercises 1 and 13, give the coordinates for point without using an new variables. 1. Kite (a, 0) 13. T 5 UV (, m) U(0, t) T(0, s) U(0, s) V(a, 0) W(0, c) V(, m) 1. Plan a coordinate proof to show that either diagonal of a parallelogram divides the parallelogram into two congruent triangles. a. Name the coordinates of parallelogram at the right. nswers ma var. Sample: (p, r); (p 1 m, r); (m, 0); (0, 0) b. What do ou need to do to show that n and n are congruent? Show that corresponding sides are. c. How will ou determine that those parts are congruent? Use the istance Formula. lassif each quadrilateral as precisel as possible. 15. (3a, 3a), (3a, 3a), (3a, 3a), (3a, 3a) square 16. (c, d 1 e), (c, d), (c, d e), (0, d) kite Prentice Hall Gold Geometr Teaching Resources opright b Pearson ducation, Inc., or its affiliates. ll Rights Reserved. 7

5 Name lass ate 6-9 Practice Form G Proofs Using oordinate Geometr Use coordinate geometr to prove each statement. Follow the outlined steps. 1. ither diagonal of a parallelogram divides the parallelogram into two congruent triangles. Given: ~ Prove: n > n a. Use the figure at the right. raw. (0, 0) (m, 0) b. Which theorem should ou use to show that n and n are congruent? plain. SSS; side lengths can be shown to be equal using coordinate geometr. c. Which formula(s) will ou need to use? istance Formula d. Show that n and n are congruent. 5. the istance Formula, both are equal to "p 1 r. 5. the istance Formula, both are equal to m. 5 b the Refleive Propert of qualit. So, the triangles are congruent b SSS.. The diagonals of a parallelogram bisect one another. Given: ~ Prove: The midpoints of the diagonals are the same. (p, r) (p m, r) a. How will ou place the parallelogram in the coordinate plane? nswers ma var. Sample: with vertices (p, r); (p 1 m, r); (m, 0); and (0, 0) b. Find the midpoints of and. What are the coordinates of the midpoints? nswers ma var. Sample: The midpoint of is ( p 1 m, r 1 m ); the midpoint of is (p, r ). c. re the midpoints the same? o the diagonals bisect one another? es; es d. Reasoning Would using a different parallelogram or labeling the vertices differentl change our answer? plain. nswers ma var. Sample: No; the midpoints would still have identical coordinates. 3. How can ou use coordinate geometr to prove that if the midpoints of a square are joined to form a quadrilateral, then the quadrilateral is a square? plain. Use the Midpoint Formula to find the midpoints of the square. Then use the Slope Formula to show that consecutive sides are perpendicular, and the istance Formula to show that all sides are congruent. Prentice Hall Gold Geometr Teaching Resources opright b Pearson ducation, Inc., or its affiliates. ll Rights Reserved. 83

6 Name lass ate 6-9 Practice (continued) Form G Proofs Using oordinate Geometr Tell whether ou can reach each conclusion below using coordinate methods. Give a reason for each answer.. triangle is isosceles. es; use the istance Formula. ou would need to prove that two sides of the triangle are congruent. ou could do this b finding the distances between the points that form the triangle. 5. The midpoint of the hpotenuse of a right triangle is equidistant from the three vertices. es; find the midpoint of the hpotenuse b using the Midpoint Formula. Then find the distance of this midpoint from each verte b using the istance Formula. 6. If the midpoints of the sides of an isosceles trapezoid are connected, the will form a parallelogram. es; find the midpoints of the sides b using the Midpoint Formula. Then use the Slope Formula to find the slopes of the segments formed b connecting these midpoints. If opposite sides are parallel, then the figure formed is a parallelogram. 7. The diagonals of a rhombus bisect one another. es; use the Midpoint Formula to find the midpoint of the diagonals. If the midpoints are the same, then the diagonals bisect one another. Use coordinate geometr to prove each statement. 8. The segments 9. The median to the 10. The segments joining joining the midpoints base of an isosceles the midpoints of a of a rhombus form a triangle is perpendicular quadrilateral form rectangle. to the base. a parallelogram. ( a, 0) (0, b) (a, 0) (0, b) (b, c) (d, e) ( a, 0) (a, 0) (0, 0) (a, 0) (0, b) The midpoints are ( a, b ), ( a, b ), (a, b ), and ( a, b ). The quadrilateral formed b these points has sides with vertical and horizontal slopes. Therefore, the consecutive sides are perpendicular, making the quadrilateral a rectangle. The median meets the base at (0, 0), the midpoint of the base. Therefore, the median has undefined slope, or is vertical. ecause the base is a horizontal segment, the median is perpendicular to the base. The midpoints are ( a, 0), (a 1 d, e ), ( b 1 d, c 1 e ), and ( b, c ). ne pair of opposite sides has a slope of e, and the d other pair has a slope of c. Therefore, it b a is a parallelogram. Prentice Hall Gold Geometr Teaching Resources opright b Pearson ducation, Inc., or its affiliates. ll Rights Reserved. 8

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