Slopes of Lines Notes What is slope? Find the slope of each line. 1
Find the slope of each line. Find the slope of the line containing the given points. 6, 2!!"#! 3, 5 4, 2!!"#! 4, 3 Find the slope of the line containing the given points. 3, 3!!"#! 4, 3 8, 3!!"#! 6, 2 5, 2!!"#! 4, 1!!, 5!!"#!!!, 3 2
Partitioning a Segment Notes Can you find the midpoint of the line segment? Now with the same line partition the segment in a ratio of 1:1. Sometimes we need to break down segments into more than just two even pieces. 1. Find the point, P, that lies along the direct line segment from A = -7 to B = 8 and partitions the segment into the ratio of 2:3. 2. Find the point, P, that lies along the direct line segment from A = 9 to B = -4 and partitions the segment into the ratio of 2:4. (use the vertical number line) 3. Find the point, P, that lies along the direct line segment from A = -6 to B = 8 and partitions the segment into the ratio of 3:2. 3
4. Given(the(points(A(03,(04)(and(B(5,(0),(find(the( coordinates(of(the(point(p(on(directed(line(segment( AB (that(partitions( AB (in(the(ratio(2:3.( 5. Given(the(points(A(8,(05)(and(B(4,(7),(find(the( coordinates(of(the(point(p(on(directed(line(segment( AB (that(partitions( AB (in(the(ratio(1:3.( 6. (Find(the(coordinates(of(point(P(that(lies(on(the(line( segment MQ,(M(09,(05),(Q(3,(5),(and(partitions(the( segment(at(a(ratio(of(2(to(5. 4
Equations of Lines Notes Slope Point Slope Form Slope Intercept Form (h, k) Form Depending upon what you are given you can use different equations for find the equation of a line. Write an equation in slope-intercept form of the line having the given slope and y-intercept.!"#$%:! 5,!!"#$%&$'#! 2!"#$%:!!, (0, 3)!! Write an equation in slope-intercept form of the line having the given slope and y-intercept.!"#$%:!12,!!"#$%&$'#!!!"#$%:!!,!: 8!! Write an equation in slope-intercept form of the line with the given slope and through the given point.! = 2, (3, 11)! =!, ( 3, 6)! Write an equation in slope-intercept form of the line with the given slope and through the given point.! = 4, ( 4, 8)! =!, ( 2, 5)! Write an equation of the line through each pair of points in slope-intercept form. 12, 6!!"#! 8, 9 0, 5!!"#! 3, 3 5
Write an equation of the line through each pair of points in slope-intercept form. 2, 4!!"#! 4, 11 3, 2!!"#! 3, 4 Find the equation of the graphed line. Find the equation of the graphed line. 6
Geometry Unit 3 Note Sheets Introduction to Parallel and Perpendicular Lines Notes Determine the slopes of the lines on each graph. 1. 2. 3. 4. 5. 6. 7. Given that each set of lines above is parallel, what can you conclude about the slopes of parallel lines? 8. What happens if a line has the same slope and the same y-intercept? 7
Geometry Unit 3 Note Sheets Determine the slopes of the lines on each graph. Then use a protractor to measure the angle of intersection. 9. 10. Angle of Intersection = 11. Angle of Intersection = 12. Angle of Intersection = 13. Angle of Intersection = 14. 15. Angle of Intersection = Angle of Intersection = What do you notice about each set of slopes in the graphs above? 16. What do you notice about the angles of intersection in the graphs above? 17. What kind of lines would you classify these as? Explain. 8
Slopes and Parallel Lines Notes What is the connection between slope and parallel lines? Slope is useful for determining whether two lines are parallel. Slope Criterion for Parallel Lines Because the theorem above has a biconditional (if and only if) you can use it in either direction. Recall the different ways to write equations of lines. Slope Intercept Form Point Slope Form (h, k) Form 9
Write an equation of the line that passes through point P and is parallel to the line with the given equation. 10
Slopes and Perpendicular Lines Notes What is the connection between slope and perpendicular lines? Slope is useful for determining whether two lines are perpendicular. Slope Criterion for Parallel Lines Because the theorem above has a biconditional (if and only if) you can use it in either direction. 11
The line with the given equation is perpendicular to line j at point R. Write an equation of line j. 1. 2. 3. 4. 5. 6. 7. 12
Perpendicular Bisector Notes Bisector - Instructions to Construct a Perpendicular Bisector 1. Place your compass point on A and stretch the compass MORE THAN half way to point B, but not beyond B. 2. With this length, swing a large arc that will go BOTH above and below!". (If you do not wish to make one large continuous arc, you may simply place one small arc above!" and one small arc below!".) 3. Without changing the span on the compass, place the compass point on B and swing the arc again. The two arcs you have created should intersect. 4. With your straightedge, connect the two points of intersection. 5. This new straight line bisects!". Label the point where the new line and!" cross as C. 1. 2. 3. 13
Construct a perpendicular bisector and then name all the relationships that we know about the figure, and what values we cannot state specific relationships about. Relationships What cannot be assumed 14