Geometry Unit 5 Geometric and Algebraic Connections Table of Contents Lesson 5 1 Lesson 5 2 Distance.p. 2-3 Midpoint p. 3-4 Partitioning a Directed Line. p. 5-6 Slope. p.7-8 Lesson 5 3 Revisit: Graphing Linear Equations...p. 8 Writing equations of Lines....p. 9 Parallel and Perpendicular p.10-11 Lesson 5 4 Lesson 5 5 Lesson 5 6 Proving Quadrilaterals p.12- Proving Triangles.. p. Area and Perimeter.. p. Graphing Circles from Standard form.. p. Converting General form to Standard form.. p. Lesson 5 7 Density... p. 1
Lesson 5 1: Distance and Midpoint Distance The tells us how far apart two points are from each other. Distance Formula: The distance between two points (x1, y1) and (x2, y2) is: Formula Sheet! Example 1: Find the length of the line shown: 1) Find the length of the line segment shown: 2) Find the length of the line segment shown: 2
Example 2: Find the distance between (3, 2) and ( 2, 4): Find the distance between the two points: 3) (5, 2) and (3, 8) 4) ( 2, 0) and ( 4, 5) 5) (7, 1) and ( 5, 3) Midpoint The midpoint is the point that lies between the two points. Formula Sheet! The midpoint formula is: Midpoint Formula: x x y y When given 1, 1 and 2, 2 the midpoint, 2 2 1 2 1 2 x y x y Example 3: Find the midpoint between (7, 1) and (5, 7). Example 4: Find the midpoint of the line segment below. 3
Find the midpoint of the ordered pairs below: 6) ( 10, 8) and (3, 1) 7) ( 7, 9) and (4, 8) 8) (0, 6) and (4, 0) Find the midpoint of the line segments below: 9) 10) Example 5: Find the other endpoint of the line segment with the given endpoint and midpoint. Endpoint: ( 10, 2) Midpoint (6, 1) YOU TRY Find the other endpoint of the line segment with the given endpoint and midpoint. 11) Endpoint ( 3, 4) and Midpoint (0, 9) 12) Endpoint: ( 6, 6) Midpoint: (10, 8) 4
A (2, 6) Lesson 5 2: Partitioning a Line Segment and Slope Directed Segment (This will help you partition a directed line segment.) A directed line segment is a line segment from one point to another point in the coordinate plane. The segment is described by an ordered pair of the directional change of x followed by the directional change of y. Find the components (vectors) of AB in each problem below. Example 6: Example 7: Example 8: B (2, 2) Point A ( 1, 3) & Point B (5, 2) B (8, 2) A ( 3, 2) Partitioning a Directed Line Segment Step 1: Determine direction that you are traveling (from what point to what point) Step 2: Find the horizontal and vertical distance from one point to the other (Keep the sign) Step 3: Multiply (ratio: part to whole) by horizontal distance and vertical distance Step 4: Add the answer for horizontal to the x1 and add the answer for the vertical to y1 creating a point between the first two points. Formula for Partitioning a Directed Line Segment: Formula Sheet! Example 9: Find the point that partitions the segment AB in a ratio of 1:3. Given: A ( 1, 4) and B (7, 8). 5
Example 10: Find the point that partitions segment BA in a 1/2 ratio. Given A (1, 2) and B (7, 8). Example 11: Draw segment AB using A ( 8, 0) and B (4, 6). Find the point that is of the distance from A to B. YOU TRY 13) Given A ( 3, 4) and B (5, 8), find the coordinates of point P on the directed line segment AB that partitions AB into a 1:3 ratio. 14) Given A (5, 1) and B (5, 9), find the coordinates of the point P on the directed line segment AB that partitions AB in the ratio of 1:3. 15) Given A ( 9, 4) and B (6, 4), find the coordinates of the point P on the directed line segment AB that partitions AB in the ratio of 2/3. 16) Given A ( 4, 8) and B ( 1, 4), find the coordinates of point R on a directed line segment BA into a 1:3 ratio 17) Given points A (1, 2) and B (6, 12), find the point that is 2/5 the distance from A to B. 6
Slope Remember, the formula for slope, or rate or change, of the line through points (x1,y1) and (x2, y2) is: Formula Sheet! Slope Formula The of a non-vertical line is the ratio of vertical change (the rise) to horizontal change (the run). Slopes can be positive, negative, undefined, or zero. Example 1: Find the slope of the line passing through ( 2, 2) and (4, 1). Also, use the graph of the coordinates to find the slope. 1) Find the slope of the line passing through (4, 7) and ( 8, 7). 2)Use the graph to determine the slope of the line. 7
Example 2: Without graphing, classify the lines as increasing, decreasing, horizontal, or vertical and explain how you know. a) (3, 4) and (1, 6) b) (2, 1) and (2, 5) c) ( 4, 5) and ( 6, 3) Revisit: Graphing Linear Equations Graph each of the following equations. Example 3: y = 3x + 5 Example 4: 2x + 3y = 6 Example 5: x = 3 Example 6: y = 5 8
Lesson 5 3: Writing Equations of Lines and Parallel/Perpendicular Lines Writing Linear Equations When an equation is written in slope intercept form, or y = mx + b, m represents the and b represents. Rewrite each of the following into slope-intercept form. Identify the slope and y-intercept. Example 7: 3x + 2y = 6 3) 4x 5y = 10 Slope y-intercept Example 8: Write an equation of the line below in slope-intercept form. Slope y-intercept 4) Write an equation of the line below in slope-intercept form. The point-slope formula can also be used to write an equation. Formula Sheet! Point-Slope Formula y y1 = m(x x1) Example 9: Write an equation of the line that passes through (2, 3) and has a slope of - 1 2. Example 10: Write an equation of the line that passes through ( 2, 1) and (3, 4). 5) Write an equation of the line that passes through ( 1, 5) and has a slope of. 6) Write an equation of the line that passes through (2, 7) and ( 4, 1). 9
Parallel and Perpendicular Lines Two lines are parallel if they have the and. Two lines are perpendicular if the slopes of the lines are slopes. Example 11: Find the equation of a line that goes that goes through the point ( 2, 0) and that is parallel to y = x + 1. Example 12: Find the equation of a line that passes through the point (1, 3) that is perpendicular to the line passing through (1, 3) and (2, 2). Example 13: Find the equation of a line that passes through the point (2, 3) that is parallel to the line passing through (3, 4) and (4, 2). Example 14: Find the equation of the line that passes through the point (1, 2) and perpendicular to the line 2x + y = 5. Example 15: Find the equation of a line that passes through ( 1, 3) that is perpendicular to the line passing through ( 1, 4) and (5, 7). Example 16: For what value of n are the lines 8x + y = 4 and nx + y = 8 parallel? Example 17: For what value of n are the lines 5x + 2y = 8 and nx + 3y = 6 perpendicular? 10
7) What is the equation of the line that goes through the point ( 1, 4) and that is parallel to y = 2x + 3? 8) What is the equation of the line that goes through the point (1, 4) and that is perpendicular to the line y = x + 2? 9) What is the equation of a line that goes through the point ( 3, 8) and is perpendicular to a line that goes through the points (0, 2) and (8, 4)? 10) What is the equation of a line that goes through the point ( 1, 7) and is parallel to a line that goes through the points ( 3, 5) and (1, 11). 11) For what value of n are the lines 3x + 2y = 8 and nx 3y = 1 perpendicular? 12) For what value of n are the lines of 5x + 3y = y and nx 2y = 8 parallel? 11
Lesson 5 4: Proving Quadrilaterals and Triangles Page 118 & 120 State Study Guide; Know how to prove a parallelogram, rectangle, rhombus, square. 12