Practice Question Which of the five points on the graph above has coordinates (x,y) such that x ϩ y ϭ 1? a. A b. B c. C d. D e. E Answer d. You must determine the coordinates of each point and then add them: A (2,Ϫ4): 2 ϩ (Ϫ4) ϭϪ2 B (Ϫ1,1): Ϫ1 ϩ 1 ϭ 0 C (Ϫ2,Ϫ4): Ϫ2 ϩ (Ϫ4) ϭϪ6 D (3,Ϫ2): 3 ϩ (Ϫ2) ϭ 1 E (4,3): 4 ϩ 3 ϭ 7 Point D is the point with coordinates (x,y) such that x ϩ y ϭ 1. Lengths of Horizontal and Vertical Segments The length of a horizontal or a vertical segment on the coordinate plane can be found by taking the absolute value of the difference between the two coordinates, which are different for the two points. A E B D 1 C 1 –GEOMETRY REVIEW– 139 Example Find the length of A ෆ B ෆ and B ෆ C ෆ . A ෆ B ෆ is parallel to the y-axis, so subtract the absolute value of the y-coordinates of its endpoints to find its length: A ෆ B ෆ ϭ |3 Ϫ (Ϫ2)| A ෆ B ෆ ϭ |3 ϩ 2| A ෆ B ෆ ϭ |5| A ෆ B ෆ ϭ 5 B ෆ C ෆ is parallel to the x-axis, so subtract the absolute value of the x-coordinates of its endpoints to find its length: B ෆ C ෆ ϭ |Ϫ3 Ϫ 3| B ෆ C ෆ ϭ |Ϫ6| B ෆ C ෆ ϭ 6 Practice Question A B C (Ϫ2,7) (Ϫ2,Ϫ6) (5,Ϫ6) A B C (Ϫ3,3) (Ϫ3,Ϫ2) (3,Ϫ2) –GEOMETRY REVIEW– 140 What is the sum of the length of A ෆ B ෆ and the length of B ෆ C ෆ ? a. 6 b. 7 c. 13 d. 16 e. 20 Answer e. A ෆ B ෆ is parallel to the y-axis, so subtract the absolute value of the y-coordinates of its endpoints to find its length: A ෆ B ෆ ϭ |7 Ϫ (Ϫ6)| A ෆ B ෆ ϭ |7 ϩ 6| A ෆ B ෆ ϭ |13| A ෆ B ෆ ϭ 13 B ෆ C ෆ is parallel to the x-axis, so subtract the absolute value of the x-coordinates of its endpoints to find its length: B ෆ C ෆ ϭ |5 Ϫ (Ϫ2)| B ෆ C ෆ ϭ |5 ϩ 2| B ෆ C ෆ ϭ |7| B ෆ C ෆ ϭ 7 Now add the two lengths: 7 ϩ 13 ϭ 20. Distance between Coordinate Points To find the distance between two points, use this variation of the Pythagorean theorem: d ϭ ͙(x 2 Ϫ x ෆ 1 ) 2 ϩ ( ෆ y 2 Ϫ y 1 ෆ ) 2 ෆ Example Find the distance between points (2,Ϫ4) and (Ϫ3,Ϫ4). C (2,4) (Ϫ3,Ϫ4) (5,Ϫ6) –GEOMETRY REVIEW– 141 The two points in this problem are (2,Ϫ4) and (Ϫ3,Ϫ4). x 1 ϭ 2 x 2 ϭϪ3 y 1 ϭϪ4 y 2 ϭϪ4 Plug in the points into the formula: d ϭ ͙(x 2 Ϫ x ෆ 1 ) 2 ϩ ( ෆ y 2 Ϫ y 1 ෆ ) 2 ෆ d ϭ ͙(Ϫ3 Ϫ ෆ 2) 2 ϩ ෆ (Ϫ4 Ϫ ෆ (Ϫ4)) ෆ 2 ෆ d ϭ ͙(Ϫ3 Ϫ ෆ 2) 2 ϩ ෆ (Ϫ4 ϩ ෆ 4) 2 ෆ d ϭ ͙(Ϫ5) 2 ෆ ϩ (0) 2 ෆ d ϭ ͙25 ෆ d ϭ 5 The distance is 5. Practice Question What is the distance between the two points shown in the figure above? a. ͙20 ෆ b. 6 c. 10 d. 2͙34 ෆ e. 4͙34 ෆ (1,Ϫ4) (Ϫ5,6) –GEOMETRY REVIEW– 142 Answer d. To find the distance between two points, use the following formula: d ϭ ͙(x 2 Ϫ x ෆ 1 ) 2 ϩ ( ෆ y 2 Ϫ y 1 ෆ ) 2 ෆ The two points in this problem are (Ϫ5,6) and (1,Ϫ4). x 1 ϭϪ5 x 2 ϭ 1 y 1 ϭ 6 y 2 ϭϪ4 Plug the points into the formula: d ϭ ͙(x 2 Ϫ x ෆ 1 ) 2 ϩ ( ෆ y 2 Ϫ y 1 ෆ ) 2 ෆ d ϭ ͙(1 Ϫ (Ϫ ෆ 5)) 2 ϩ ෆ (Ϫ4 Ϫ ෆ 6) 2 ෆ d ϭ ͙(1 ϩ 5 ෆ ) 2 ϩ (Ϫ ෆ 10) 2 ෆ d ϭ ͙(6) 2 ϩ ෆ (Ϫ10) ෆ 2 ෆ d ϭ ͙36 ϩ 1 ෆ 00 ෆ d ϭ ͙136 ෆ d ϭ ͙4 ϫ 34 ෆ d ϭ ͙34 ෆ The distance is 2͙34 ෆ . Midpoint A midpoint is the point at the exact middle of a line segment. To find the midpoint of a segment on the coordi- nate plane, use the following formulas: Midpoint x ϭ ᎏ x 1 ϩ 2 x 2 ᎏ Midpoint y ϭ ᎏ y 1 ϩ 2 y 2 ᎏ Example Find the midpoint of A ෆ B ෆ . B A Midpoint (5,Ϫ5) (Ϫ3,5) –GEOMETRY REVIEW– 143 Midpoint x ϭ ᎏ x 1 ϩ 2 x 2 ᎏ ϭ ᎏ Ϫ3 2 ϩ 5 ᎏ ϭ ᎏ 2 2 ᎏ ϭ 1 Midpoint y ϭ ᎏ y 1 ϩ 2 y 2 ᎏ ϭ ᎏ 5 ϩ 2 (Ϫ5) ᎏ ϭ ᎏ 0 2 ᎏ ϭ 0 Therefore, the midpoint of A ෆ B ෆ is (1,0). Slope The slope of a line measures its steepness. Slope is found by calculating the ratio of the change in y-coordinates of any two points on the line, over the change of the corresponding x-coordinates: slope ϭ ᎏ ho v r e i r z t o ic n a t l a c l h c a h n a g n e ge ᎏ ϭ ᎏ x y 2 2 Ϫ Ϫ y x 1 1 ᎏ Example Find the slope of a line containing the points (1,3) and (Ϫ3,Ϫ2). Slope ϭ ᎏ x y 2 2 Ϫ Ϫ y x 1 1 ᎏ ϭ ᎏ 3 1 Ϫ Ϫ ( ( Ϫ Ϫ 2 3 ) ) ᎏ ϭ ᎏ 3 1 ϩ ϩ 2 3 ᎏ ϭ ᎏ 5 4 ᎏ Therefore, the slope of the line is ᎏ 5 4 ᎏ . Practice Question (5,6) (1,3) (1,3) (Ϫ3,Ϫ2) –GEOMETRY REVIEW– 144 . distance between the two points shown in the figure above? a. ͙20 ෆ b. 6 c. 10 d. 2͙34 ෆ e. 4͙34 ෆ (1,Ϫ4) (Ϫ5,6) –GEOMETRY REVIEW 142 Answer d. To find the distance between two points, use the following. ( ෆ y 2 Ϫ y 1 ෆ ) 2 ෆ d ϭ ͙(1 Ϫ (Ϫ ෆ 5)) 2 ϩ ෆ (Ϫ4 Ϫ ෆ 6) 2 ෆ d ϭ ͙(1 ϩ 5 ෆ ) 2 ϩ (Ϫ ෆ 10) 2 ෆ d ϭ ͙(6) 2 ϩ ෆ ( 10) ෆ 2 ෆ d ϭ ͙36 ϩ 1 ෆ 00 ෆ d ϭ ͙136 ෆ d ϭ ͙4 ϫ 34 ෆ d ϭ ͙34 ෆ The distance is 2͙34 ෆ . Midpoint A. y 1 ෆ ) 2 ෆ Example Find the distance between points (2,Ϫ4) and (Ϫ3,Ϫ4). C (2,4) (Ϫ3,Ϫ4) (5,Ϫ6) –GEOMETRY REVIEW 141 The two points in this problem are (2,Ϫ4) and (Ϫ3,Ϫ4). x 1 ϭ 2 x 2 ϭϪ3 y 1 ϭϪ4 y 2 ϭϪ4 Plug