Line segments in a coordinate plane can be analyzed by finding various characteristics of the line including slope, length, and midpoint. These values are useful in applications and coordinate proofs. Slope = slopes are equal. slopes are negative reciprocals. Some lines are neither parallel nor perpendicular. Length of a segment is the distance between x1, y1 and x2, y 2. Distance formula Midpoint of a segment between x1, y1 and x2, y 2 Examples: Midpoint formula 1. Given A(9, -5) and B(-6, 12) a. Find the slope. b. Find AB. c. Find Q, the midpoint of AB. d. Find the equation of the line that passes through AB. 2. Given R(-2, 5) and T(4, 1) a. Find the slope. b. Find RT. c. Find S, the midpoint of RT. d. Find the equation of the line that passes through RT. 2012, TESCCC 05/16/12 page 1 of 6
Comparing Parallelism in Euclidean and non-euclidean Geometries Parallel Postulate (Euclidean ) If the Parallel Postulate is considered false, then one of the following assumptions must be considered true. Assumption 1: Through a given point not on a line, there are no lines parallel to the given line. Assumption 2: Through a given point not on a line, there is more than one line parallel to the given line. Assumption 1 applies to. Assume that the following figure is a sphere. Lines are represented by curves on the great circles of the sphere. This is a sphere and in spherical geometry, great circles represent lines. Note that all great circles intersect. Therefore there are no parallel lines. Assumption 2 applies to. Assume that the following figure is a concave disk. Lines appear to be curves on the surface of the disk, although if flattened into a plane, they would be linear. This is not a sphere, but is the inside of a curved disk. Note that p is parallel to s, t, and v, since they do not intersect. There are an infinite number of curves that could be drawn through the point and not intersect p. p s t v 2012, TESCCC 05/16/12 page 2 of 6
Guided Practice 1. Use the coordinate plane to answer questions a f. a. Given A(0, -8), B(5, 9), C(7, -6), D(-8, 6), E(3, 0), F(-4, -2), G(-6, 2), label the points on the graph. b. Find the slope of BE and EG. What does the slope indicate about these lines? c. Find the slope of FE and AC. What does the slope indicate about these lines? d. Find the length of AE. e. Find the midpoint of CF. f. Find the length and midpoint of CD. 2012, TESCCC 05/16/12 page 3 of 6
Practice Problems 1. A square is represented by the points A(1, -2), B(-3, 2), C(-7, -2), and D(-3, -6). a. Find the equation of the line passing through each side of square ABCD. b. Find the slope of each side of square ABCD. c. Determine which sides are perpendicular and which sides are parallel. d. Calculate the length of each line segment that makes up square ABCD. e. Construct a graph as an alternate representation to verify your conclusions. 2012, TESCCC 05/16/12 page 4 of 6
Use the coordinate plane to answer the questions below. 2. Given H(-1, -3), I(0, 6), J(6, -3), K(-6, 0), L(-4, 4), M(1, 2), N(3, 7), label the points on the graph. 3. Find the slope of MN and LM. What does the slope indicate about these lines? 4. Find the slope of HK and MJ. What does the slope indicate about these lines? 5. Find the length of KI. 6. Find the midpoint of KM. 7. Find the length and midpoint of JL. 2012, TESCCC 05/16/12 page 5 of 6
Euclidean versus non-euclidean Geometries 8. What comprises the surface of a sphere? 9. What is a line on a sphere? 10. Does the Parallel Postulate in Euclidean geometry hold in spherical geometry? Explain your reasoning. 11. What comprises a surface in hyperbolic geometry? 12. What is a line in hyperbolic geometry? 13. Does the Parallel Postulate in Euclidean geometry hold in hyperbolic geometry? Explain your reasoning. 2012, TESCCC 05/16/12 page 6 of 6