Distance. Warm Ups. Learning Objectives I can find the distance between two points. Football Problem: Bailey. Watson

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1 Distance Warm Ups Learning Objectives I can find the distance between two points. Football Problem: Bailey Watson. Find the distance between the points (, ) and (4, 5). + 4 = c = c 5 = c 5 = c. Using the same method on your graph paper, find the distance between the points (-, ) and (4, -) = c = c 5 = c 7. = c

2 Is there any way that we can get the x and y values without drawing the triangle on the graph??? Distance formula: d = ( y y ) + ( x x ). Find the distance between the points (, 5) and (-, 9) without drawing it on the graph. + 4 = c = c 5 = c 5 = c 4. Find the distance between the points (9, -) and (5, -7) = c = c 5 = c 7. = c 5. Find the distance between the points (4, ) and (0, 8) = c = c 6 = c 7.8 = c Find the perimeter of each shape. 6. Perimeter: 7. Perimeter: 0

3 Find the perimeter of the triangles. 8. The points are A (-4, ), B (4, 7), C (8, -) AB: = c = c 80 = c 8.9 = c BC: = c 8.9 = c AC: + 4 = c = c 60 = c.6 = c Perimeter = The points are A (-4, ), B (-4, -), C (5, -) AB = 6 BC : 9 + = c = c 85 = c 9. = c = c = c 97 = c 9.8 = c Perimeter = 5 0. Success Criteria Find the perimeter of the triangle that has the points A (, 4), B (-, 7) and C (5, -). AB: BC : AC : = c = c 4 = c 6.4 = c 4 + = c = c 5 = c 5 = c = c = c 8 = c. = c Perimeter =.7. Find the distance between the two points. a. b. + 4 = c = c 5 = c 5 = c + 7 = c = c 5 = c 7. = c. Find the distance between the points (4, -) and (6, 5) = c = c 49 = c. = c

4 . Find the distance between the points (-, 6) and (, -6) = c = c 49 = c. = c 4. Find the perimeter of triangle with coordinates A (-, -5), B (-, ) and C (5, -). AB + 7 = c = c 58 = c 7.6 = c AC = c = c 65 = c 8. = c BC 6 + = c = c 7 = c 5. = c Perimeter = Find the perimeter of a triangle with coordinates A (4, ), B (, -) and C (8,0). AB + 4 = c = c 0 = c 4.5 = c AC 4 + = c = c 0 = c 4.5 = c BC 6 + = c = c 40 = c 6. = c Perimeter = Find the perimeter of a triangle with coordinates X (7, ), Y (-, 6) and Z (-, -). XY 8 + = c = c 7 = c 8.5 = c YZ + 9 = c = c 85 = c 9. = c XZ = c = c 6 = c.7 = c Perimeter = 9.4 4

5 Slope and Equations of Lines Warm Ups Learning Objectives I can determine slopes using the formula. y y Slope Formula: m = x x. Find the slope of the line. rise m = = run 5 4 Types of Slopes Positive Negative Zero Undefined. Find the slope of the line with points A (5,) and B (9,-5). y y 5 8 m x x = = = = 5

6 Remember... What is Point-Slope Form? y y = m( x x ) What is Slope-Intercept Form? y = mx + b. Write an equation in slope-intercept form 4. Find the equation of the line. of the line with a slope of 5 and a y = x + 4 y-intercept of -4. Then graph the equation. y = 5x 4 5. Write the equation of a line in point-slope 6. Success Criteria: Write an equation of form of a line with a slope of - that the line in point-slope form that contains the point (-,-). Then graph the line. contains the points (-7,4) and (9,-4). y y y + = ( x + ) m = = = = x x y 4 = ( x + 7) 6

7 7. Write the equation in slope-intercept form of the line having the given slope and the y- intercept or points. Then graph the line a. m = -5, y-intercept = - b. m = -/4, y-int = (0, 4) y = 5x y = x Write an equation in point slope form of the line having the given slope that contains the given point. Then graph the line. a. m =, (, ) b. m = 5/7, (-, -5) y = ( x ) 5 y + 5 = ( x + ) 7 7

8 9. Write an equation of the line through each pair of points in point-slope form. a. (-, -4) and (, -4) y = 4 b. (, -) and (, 6) x = c. (-, -) and (4, 5) y y m x x = = = = y + = ( x + ) d. (4, 5) and (6, -) y y 5 6 m = = = = x x 6 4 y 5 = ( x 4) e. (, -) and (8, -) y y m = = = x x 8 7 y + = ( x ) 7 f. (-, 5) and (6, -4) y y m x x 6 9 = = = = y 5 = ( x + ) 8

9 Graphing Exploration Day Warm Ups Learning Objectives I can use Desmos to draw geometric shapes.. Using the points: (, ), (7, ), (, 4), (7, 4) a. Plot the points on a graph of this rectangle. b. Find the equations for each of the lines. Type in the equations to see that they go through the correct points. c. Restrict the values so segments connect each of the points. d. Write the equations that you used below. y = {<x<7} x = 7{<y<4} y = 4{<x<7} x = {<y<4} e. Find the distances of each side. The distances are:, 4,, 4. Using the points: (, 4), (, 7), (4, 6), (5, 9) a. Plot the points on a graph of this parallelogram. b. Find the equations for each of the lines. Type in the equations to see that they go through the correct points. c. Restrict the values so segments connect each of the points. d. Write the equations that you used below. y 4 = ( x ){ < x < }, y 4 = ( x ){ < x < 4} y 7 = ( x ){ < x < 5}, y 6 = ( x 4){4 < x < 5} e. Find the distances of each side. 9

10 Distances are:.6,.6,.6 and.6. Using the points: (, ), (5, ), (7, 7), and (, 5) a. Plot the points on a graph of this rhombus. b. Find the equations for each of the lines. Type in the equations to see that they go through the correct points. c. Restrict the values so segments connect each of the points. d. Write the equations that you used below. y = ( x ){ < x < }, y 5 = ( x ){ < x < 7} y = ( x ){ < x < 5}, y = ( x 5){5 < x < 7} e. Find the distances of each side. Each side is Create a picture with at least 4 lines in it. a. Draw the picture below b. Write the equation to the each line below. 0

11 Midpoint Warm Ups Learning Objectives I can determine midpoints using the formula. I can find the coordinate of an endpoint given a midpoint and the other endpoint. I can use algebra to solve word problems based on my understanding of a midpoint. Average The middle of a set of numbers Midpoint the point halfway between two points Formula (in the coordinate plane) x + x, y + y. ( 4, 4),(5, ) (,5),(8,) x + x y + y x + x y + y, , , (0.5, ) (.5,.5). (7, -4) (, -) x + x y + y (5, ) 4. Where should they meet? Techniques to solve a midpoint problem:. Starbucks 5. What is the other endpoint? (7,) 6. Given B lies between A and C, find C s coordinate. A (9, -) B (5, ) C(, 8) Bisect a segment, line, or plane that intersects a segment at a midpoint Congruent The same measure

12 7. Draw one diagram for this situationj. Let F be the midpoint of AH. Let C be the midpoint of AF. 8. M is the midpoint of LN. Find the value of x. 4x 7 = x + 9 x 7 = 9 x = 6 x = 9. Suppose V is the midpoint of UW. VW = 4a + and UV = 5 a. Find the length of UW. Step : Draw Step : Set up equation Step : Solve Step 4: Find the length 4a + = 5 a 6a + = 5 6a = a = UW = + = Success Criteria 0. Find the coordinate of P, the midpoint of MT, given M (6, -) and T (, 4). x + x y + y (4, 0.5). Find the length of XZ given Y is the midpoint and XY = x 6 and YZ = x a + = 5 a 6a + = 5 6a = a = UW = + =. Use the number line to find the coordinate (number) of the midpoint of each segment. a. - b. 4 c. - d. 5 e. -8 f. 0.5

13 Find the coordinates of the midpoint of a segment with the given endpoints.. 4. K( 9, ), H(5, 7) 5. W(, 7), T( 8, 4) (-, -0.5) (-, 5) (-0, -5.5) Find the coordinates of the missing endpoint if E is the midpoint of DF. 6. F(5, 8), E(4, ) 7. F(, 9), E(, 6) 8. D(, 8), E(, ) (, -) (-4, ) (5, 4) 9. Find the coordinates of B if B is the midpoint of and C is the midpoint of. To find point C: x + x y + y C(,) To find point B: x + x y + y B(, ) Suppose M is the midpoint of FG. Use the given information to find the missing information. 0. FM = x 4, MG = 5x 6, FG =?. FM = 8a +, FG = 4, a =? x 4 = 5x -6 8a + = = x 8a = 0 = x a =.5 () 4 = 9 FG = 58

14 Equations of Parallel and Perpendicular Lines Warm Ups Learning Objectives I can identify parallel and perpendicular lines based on their slopes. I can write the equation of a parallel line given a point not on the line. I can write the equation of a perpendicular line given a point not on the line. Forms of a Line Standard: Ax + Bx = C Slope-Intercept: y = mx + b Point Slope: y y = m( x x ) Discuss with your neighbors... How do we know if lines are parallel? The slopes are equal. How do we know if lines are perpendicular? The slopes are negative reciprocals.. Write the slope of the line parallel and perpendicular to each line: y = x y = -/x + Parallel: m = m = -/ Perpendicular: m = -/ m = 4

15 . Determine whether line WX and line YZ are perpendicular. Slopes: YZ- m = 4/5 WX- m = -/5 No. These slopes are not negative reciprocals.. Write the equation of the line that is parallel to the line y = -x + 5 and goes through the point (, 4). m = - point (, 4) y 4 = ( x ) 4. Write the equation of the line that is perpendicular to the line y = -x + 5 and goes through the point (, 4). m = / point (, 4) y 4 = ( x ) 5. Write the equation of the line that is parallel to the line y - = /4(x + ) and goes through the point (-, 8). m = /4 point (-, 8) y 8 = ( x + ) 4 6. Success Criteria: Given the following coordinates, A(, -) B(, -6) C(4, -) D(6, 4) Find the slope of: AB AC CD BD m = -/ m = / m = 5/ m = 5/ Which of the segments are parallel and which are perpendicular? AB and AC are perpendicular. 5

16 7. Find the equation of the line that has the following conditions. Then graph it. a. Parallel to y = x + 4, but passes through (, -) Slope: m = Point: (, -) y + = ( x ) b. Perpendicular to y = x - 5, but passes through (, 4) Slope: m = -/ Point: (, 4) y 4 = ( x ) c. Parallel to y = (x - ) +, but passes through (4, 5) Slope: m = Point: (4, 5) y 5 = ( x 4) 6

17 d. Perpendicular to y = -(x + ) - 4, but passes through (-, -) Slope: m = / Point: (-, -) y + = ( x + ) 8. Find the slopes of AB and CD. Determine if AB and CD is parallel, perpendicular, or neither. a. A(, ), B(5, 9), C(8, ), D(7, ) y y 9 6 AB: m = = = = x x 5 b. A(-, 5), B(4, 9), C(, ), D(5, 5) y y CD: m = = = = x x 7 8 Neither y y y y 5 AB: m = = = = CD: m = = = Parallel x x 4 6 x x 5 9. Find the equation of line SA in each of the following situations. a. Equation: b. Equation: Slope of SY = 5/4 Slope of SY = -/ Slope of SA = -4/5 Slope of SA = Point SA = (, 5) Point SA = (-, ) 4 y 5 = ( x ) y = ( x + ) 5 7

18 Graphing Exploration Day Warm Ups Learning Objectives I can use graph to draw geometric shapes.. Using the points: A(, ), B(4, 9), C(8, 4) a. Plot the points on a graph of this triangle. b. Find the equations for each of the lines. Type in the equations to see that they go through the correct points. c. Restrict the values so segments connect each of the points. d. Write the equations that you used below. 5 y = ( x ){ < x < 8} y 9 = ( x 4){4 < x < 8} 5 4 y = 8( x ){ < x < 4} e. Find the midpoint of segment AB and segment BC. AB: (.5, 5) BC: (6, 6.5) f. Find the equation of the line segment that goes from the midpoint of AB and the midpoint of BC. Draw this on the graph. y y m = = = = y 5 = ( x.5){.5 < x < 6} x x g. What do you notice about this line segment? It is parallel to one of the segments. It is also half of its length. 8

19 . Using the points: A(4, 5), B(7, 6), C(, ), D(9, ) a. Plot the points on a graph of this trapezoid. b. Find the equations for each of the lines. Type in the equations to see that they go through the correct points. c. Restrict the values so segments connect each of the points. d. Write the equations that you used below. y = ( x ){ < x < 9} y = 4( x ){ < x < 4} y 6 = ( x 7){7 < x < 9} y 5 = ( x 4){4 < x < 7} e. Find the midpoint of segment AC and segment BD. AC: (.5, ) BD: (8, 4.5) f. Find the equation of the line segment that goes from the midpoint of AC and the midpoint of BD. Draw this on the graph. y y m = = = = y = ( x.5){.5 < x < 8} x x g. What do you notice about this line segment? It is parallel to the third side and the average of its length.. Create your own problem. a. Pick and 4 points and graph them. b. Find the equations that connect the points and restrict the values so you can graph the segments. c. Find the midpoint of each segment. d. Find the equations that connect all of the midpoints with line segments. Write those equations below. e. Find the midpoints for each segment in section d. f. Write the equations of the lines that connect all of the midpoints that you found in part e. g. What do you notice about the slopes of the lines? What shape did this produce? 9

20 Perpendicular Bisectors Warm Ups Learning Objectives I can write the equation of the perpendicular bisectors of a segment. I can determine midpoints using the formula. Review: Find the midpoint between the points (5, -) and (-, 6). What does it mean to be a perpendicular bisector? Sketch perpendicular bisectors to segments AB and CD.. Sketch the perpendicular bisectors on the graphs below. 0

21 . Find the equation of a perpendicular bisector of a segment with the endpoints (, 6) and (5, ) (.5, 9) 6 6 m = = = 5 m = -/ y 9 = ( x.5). Find the equation of a perpendicular bisector of a segment with the endpoints (-, -4) and (9, 6) m = = = (9,6) m = - y 6 = ( x 9) 4. Find the equation of a perpendicular bisector of a segment with the endpoints (, 7) and (-, 4). Midpoint Slope Perp Slope Equation (, 5.5) 4 7 m = = = 4 4 m = -4/ 4 y 7 = ( x ) 5. Success Criteria: Find the equation of a perpendicular bisector of a segment with the endpoints (8, ) and (5, -). Midpoint Slope Perp Slope Equation (6.5, 0.5) m = = = 5 8 m = - y = ( x 8)

22 6. Draw the perpendicular bisector on the graphs. a. b. 7. Write the equation of the perpendiculars that you drew in the previous question. a. Equation : _ y.5 = ( x 5) b. Equation: y 4 = ( x ) 8. Write the equations for the perpendicular bisectors on the graphs below. a. b. y = ( x ) y 7.5 = 8( x + ) 9. Find the equation of the line that is the perpendicular bisector of PQ for the given endpoints. a. P(5, ) Q(7, 4) Midpoint Slope Perp Slope Equation m = = = 7 5 (6,) m = - y = ( x 6) b. P(-, 9) Q(-, 5) Midpoint Slope Perp Slope Equation (,7) m = = = m = / y 7 = ( x + )

23 c. P(-6, -) Q(8, 7) Midpoint Slope Perp Slope Equation (,) m = = = m = -7/4 7 y = ( x ) 4 d. P(-, ) Q(0, -) Midpoint Slope Perp Slope Equation (, ) m 4 = = = 0 m = / y + = ( x + ) e. P(0,.6) Q(0.5,.) Midpoint Slope Perp Slope Equation (0.5,.85)..6.5 m = = = m = - y +.85 = ( x 0.5)

24 Review Warm Ups. Find the slopes of line AB if A(, ) and B(6, -). m = = = 6. Find the slopes of line AB if A(8, ) and B(-, 5). 5 m = = 8 0. Find the slopes of the following graphs. 6 4 (a) m = = = 6 4 (b) 7 9 m = = Given the points T(, -), R(4, 4), L(, -), P(6, -), determine whether lines CS and KP are parallel, perpendicular or neither. Show work to support your answer. (a) Find the slope of TR. 4 6 m = = = 4 (b) Find the slope of LP. m = = = 6 4 (c) Determine if they are parallel, perpendicular, or neither. Perpendicular 4

25 5. Refer to the points on the graph below. Determine whether segments QW and PY are parallel, perpendicular or neither. Include the reasoning to support your answer. (a) Find the slope of QW. m = 6/4 = / (b) Find the slope of PY. m = 6/4 = / (c) Determine whether QW and PY are parallel, perpendicular, or neither. Parallel (d) Find the distance of YP = c = c 5 = c 7. = c (e) Find the distance of WY = c = c 5 = c 7. = c 6. Refer to the points on the graph below. Determine whether segments QW and PY are parallel, perpendicular or neither. Include the reasoning to support your answer. (a) Find the slope of QW. m = -5/5 = - (b) Find the slope of PY. m = 6/6 = (c) Determine whether QW and PY are parallel, perpendicular, or neither. Perpendicular 5

26 7. Write the equation of a line that has a slope of and passes through the point (-, 9). y 9 = (x + ) 8. Write the equation of the line that passes through the points (, -) and (5, -6). (a) Find the slope. m 6 4 = = = 5 (b) Write the equation. y + = -(x ) 9. Write the equation of the line that passes through the points (-5, -) and (6, ). (a) Find the slope. 5 m = = 6 5 (b) Write the equation. 5 y + = ( x + 5) 0. A line PQ has a slope of -/7. What is the slope of the line that is: (a) parallel to PQ m = -/7 (b) perpendicular to PQ m = 7/ =.5. In the diagram, PO and LO meet at 90 o. If P(, 9) and O(6, 4): (a) Find the slope of PO m = = 6 (b) Find the slope of LO m = 5 (c) Write the equation for LO y 4 = /5(x 6). Write an equation of a line passes through point (, 4) and is parallel to the line y = 5x +. y 4 = -/5(x ) 6

27 . Write an equation of a line passes through point (, 4) and is perpendicular to the line y = 5x +. y 4 = /5(x ) 4. Find the midpoint between the points R(, 0) and S(5, -) = (4, 4) 5. Find the equation for the perpendicular bisector of the segment PY shown on the graph. (a) Find the midpoint. (0.5, 4) (b) Find the slope of PY m = 4/5 (c) Find the perpendicular slope m = -5/4 (d) Write the equation of the perpendicular bisector. y 4 = -5/4(x -.5) 6. A triangle has points at A(, 5), B(7, 9) and C(-4, 8). Graph the triangle below. Find the perimeter of this triangle. AB = c = c = c 5.65 = c AC 7 + = c = c 58 = c 7.6 = c BC + = c + = c = c.04 = c Perimeter = A triangle has points at A(-, ), B(5, -) and C(7, ). Graph the triangle below. Find the perimeter of this triangle. AB 8 + = c = c 68 = c 8. = c AC 0 Perimeter = BC + = c = c 8 = c.8 = c 7