Geometry R. Unit 12 Coordinate Geometry. Day Classwork Day Homework Wednesday 3/7 Thursday 3/8 Friday 3/9

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1 Geometry R Unit 12 Coordinate Geometry Day Classwork Day Homework Wednesday 3/7 Thursday 3/8 Friday 3/9 Unit 11 Test Review Equations of Lines 1 HW 12.1 Perimeter and Area of Triangles in the Coordinate Plane 2 HW 12.2 Monday 3/12 Perimeters and Areas of Polygonal Regions in the Coordinate Plane Unit 12 Quiz 1 3 HW 12.3 Tuesday 3/13 Wednesday 3/14 Thursday 3/15 Friday 3/16 Dilations of Lines 4 HW 12.4 Searching a Region in a Plane (d=rt) Unit 12 Quiz 2 5 HW 12.5 Parallel and Perpendicular Lines 6 HW 12.6 No School Monday 3/19 Tuesday 3/20 Rotating 90 about a Point Unit 12 Quiz 3 Equations of Special Segments and Finding Triangle Centers in the Coordinate Plane 7 HW HW 12.8 Wednesday 3/21 Dividing Segments Proportionally 9 HW 12.9 Thursday 3/22 Review Unit 12 Quiz 4 10 Review Sheet Friday 3/23 Monday 3/26 Tuesday 3/27 Review 11 Review Sheet Review 12 Review Sheet Unit 12 Test 13

2 Review of Writing Equations of Lines [1] Slope-Intercept Form of a Line Point-Slope Form of a Line 1. Write the equation of line with a slope of 2 and passing through the point (-1, 5) 2. Write the equation of line passing through the points (4, -3) and (5, 11). 3. Write the equation of a line parallel to the x-axis, passing through the point (8, -2). 4. Write the equation of a line parallel to the y-axis, passing through the point (-2, 5).

3 Consider the rectangular region: 1. Decide if a line with a slope given, passing through the origin will intersect this rectangular region. If so, which boundary points of the rectangle does it intersect? i. m 2 ii. 1 m 2 iii. 1 m 3 b. What is the slope of a line that passes through the origin and the lower right vertex of the rectangle? c. What is the slope of a line that passes through the origin and the upper left vertex of the rectangle? d. For which values of m would a line of slope m through the origin intersect this rectangular region?

4 2. Consider the triangular region in the plane given by the triangle with vertices A(0, 0), B(2, 6), and C(4, 2). a. The horizontal line y = 2 intersects this region. What are the coordinates of the two boundary points it intersects? What is the length of the horizontal segment within the region along this line? A B C b. Graph the line 3x y 5. Find the points of intersection with the triangular region and label them as X and Y. c. Find the length of segment XY to the nearest tenth. d. A robot starts at position B(2, 6) and moves vertically downward towards the xaxis at a constant speed of 0. 2 units per second. When will it hit the lower boundary of the triangular region that falls in its vertical path?

3. Graph the rectangular region, A(-1, 4), B(4, 4), C(4, 1), and D(-1, 1): a. Draw lines that pass through the origin and through each of the vertices of the rectangular region.

5 3. Graph the rectangular region, A(-1, 4), B(4, 4), C(4, 1), and D(-1, 1): a. Draw lines that pass through the origin and through each of the vertices of the rectangular region. Do each of the four lines cross multiple points in the region? Explain. b. A robot is positioned at D and begins to move in a straight line with slope m = 1. When it intersects with a boundary, it then reorients itself and begins to move in a straight line with a slope of m = 1. What is the location of the next intersection the robot makes with the 2 boundary of the rectangular region? c. What is the approximate distance of the robot s path in part b?

6 1. Find the perimeter of OAB : Perimeter and Area of Triangles in the Coordinate Plane [2] 2. Find the area of the shaded region: a. b. To the nearest tenth c. Label points (3,1) and (6,4) and find the area of the triangle

3. a. Find the area of the triangle by boxing in the triangle: b. Using the same procedure as part a, develop a general formula for the area of the triangle below. C i.

7 3. a. Find the area of the triangle by boxing in the triangle: b. Using the same procedure as part a, develop a general formula for the area of the triangle below. C i. Find the coordinates of A, B, and C in terms of x1, x2, y1, and y2 ii. Find the area of the rectangle iii. Find the area of each triangle iv. Find an expression for the area of OAB

8 General Formula for Area of Triangle with one Vertex at (0, 0) 1 A ( x1 y2 x2 y1 ) 2 Find the area of the triangles with the vertices listed using one of the formulas above. 4. O(0, 0), A(5, 6), B(4, 1) 5. O(0, 0), A(3, 2), B(-2, 6) 6. In question #3, we saw that the area of the triangle with vertices O(0, 0), A(5, 2), and B(3, 4) had an area of 7 square units. Can you find the area of a triangle with vertices O (10, 12), A (15, 10), and B (13, 8)? [Hint: find a transformation that maps the preimage to the image] 7. a. Write the numerical coordinates of points A, B, and C b. Identify a translation that takes C to C (0, 0). c. Apply this translation on points A and B. Tell the coordinates of A and B. d. Use the formula above to find the area of the triangle

This process of translating a figure and then applying the area

x y x y y x y x y x We call this the shoelace formula: 8.

use shoelace method to check yourself): 9.

9 This process of translating a figure and then applying the area formula can be generalized as: 1 ( ) x y x y x y y x y x y x We call this the shoelace formula: 8. Find the area of ABC by translating to the origin (feel free to use shoelace method to check yourself): 9. Find the area of ABC by translating to the origin (feel free to use shoelace method to check yourself):

10 Perimeter and Area of Polygonal Regions in the Coordinate Plane [3] 1. Find the area and perimeter of the shaded figure: Method 1: Boxing In Method 2: Shoelace Theorem or General Formula for a Triangle Draw diagonal(s) to create triangles 2. Find the area of the figure below using two different methods:

11 3. A quadrilateral region is defined by the system of inequalities below: y x + 6 y 2x + 12 y 2x 4 y x + 2 a. Sketch the region. b. Determine the vertices of the quadrilateral. c. Find the perimeter of the quadrilateral region. d. Find the area of the quadrilateral region.

4. A quadrilateral region is defined by the system of inequalities below: y x + 5 y x 4 y 4 y 5 x 4 4 a. Sketch the region. b. Determine the vertices of the quadrilateral.

12 4. A quadrilateral region is defined by the system of inequalities below: y x + 5 y x 4 y 4 y 5 x 4 4 a. Sketch the region. b. Determine the vertices of the quadrilateral. c. Find the perimeter of the quadrilateral region. d. Find the area of the quadrilateral region.

13 Dilations of Lines in the Coordinate Plane [4] 1. Find the equation of the line y = 2x + 2 after a dilation with a scale factor of 3 and centered at the origin. Steps to write the equation of the line after a dilation center at origin 1. Pick 2 points from the line. 2. Dilate each point using a scale factor of Plot new points on the graph and connect to form a line. 4. Find the slope of the new line. 5. Find the y-intercept of the new line. 6. Write the equation of the new line. 2. The line y = 2x + 2 is dilated by a scale factor of 2 and centered at the origin. Write the equation of the new line after the dilation.

14 3. The line is dilated by a scale factor of and centered at the origin. Which equation represents the image of the line after the dilation? 1) 2) 3) 4) 4. Line is transformed by a dilation with a scale factor of 2 and centered at the origin. Find the equation of the line after the dilation. SUMMARY: A dilation takes a line to a parallel line if the is NOT on the line. Slope stays the. y intercept is multiplied by. 5. The line y = 3x is dilated by a scale factor of 2 and centered at the origin. Write the equation that represents the image of the line after the dilation. SUMMARY: A line remains if the center of dilation is on the line.

15 SEARCHING A REGION IN A PLANE [5] Slope Positive Negative Zero Undefined Slope is used to describe the measurement of steepness of a straight line. Slope is also described as a. Slope is a ratio that can be expressed as: m = rise run or change in y change in x or vert change horiz change or y 2 y 1 x 2 x 1 Pythagorean Theorem Distance Formula Used to find the missing side in a right triangle Used to find the distance between 2 points R Distance = Rate x Time Ex) If a car travels for 3 hours at 60 mph, how far has the car traveled? R Time = Distance Rate Ex) How long will it take a car traveling at 60 mph to go 300 miles? R Rate = Distance Time Ex) What is a car s average rate of speed if it has traveled 220 miles in 4 hours?

16 Consider the following: Students in a robotics class must program a robot to move about an empty rectangular warehouse. The program specifies location at a given time, t, seconds. The room is twice as long as it is wide. Locations are represented as points in a coordinate plane with the southwest corner of the room deemed the origin, (0,0), and the northeast corner deemed the point (2000 ft., 1000 ft. ), as shown in the diagram below. (2000 ft.,1000 ft. View of the Warehouse (0,0) The first program written has the robot moving at a constant speed in a straight line. At time t = 1 second, the robot is at position (30, 45), and at t = 3 seconds, it is at position (50, 75). Answer the questions below to program the robot s motion. 1. Plot the given points on the smaller graph at the right. 2. Draw a line connecting the points. 3. How much did the x-coordinate change in 2 secs? In 1 sec? 4. How much did the y-coordinate change in 2 secs? In 1 sec? 5. At what time did the robot start its motion? 6. Where did the robot start its motion? 7. What is the ratio of change in y to change in x? What does this rate of change represent? 8. What is the equation of the line of motion? 9. To the nearest foot, how far did the robot travel in 2 seconds? 10. What is the robot s rate of speed?

17 11. Which wall of the room will the robot hit? (2000 ft.,1000 ft. ) (0,0) 12. Where is the location of impact (i.e. where will the robot hit the wall)? 13. What is the distance to the wall? 14. At what time will the robot hit the wall to the nearest tenth of a second?

18 PARALLEL AND PERPENDICULAR LINES [6] The lines l 1 and l 2 are perpendicular if and only if their slopes are of each other. Ex: Find the slope of a line perpendicular to y 2x 5 Slopes of Perpendicular Lines (1, m 1 ) (1, m 2 ) l 1 l 2 1) Write an equation of the line that passes through the point (7, -3) that intersects the line 2x + 5y = 7 to form a right angle. 2) Determine whether the lines given by the equations 2x + 3y = 6 and y = 3 x + 4 are 2 perpendicular. Support your answer. 3) Two lines having the same y-intercept are perpendicular. If the equation of one of these lines is y = 4 x + 6, what is the equation of the second line? 5 4) What is the relationship between two coplanar lines that are perpendicular to the same line?

19 5) Given two lines, l 1 and l 2, with equal slopes and a line k that is perpendicular to one of these two parallel lines, l 1 : a) What is the relationship between line k and the other line, l 2? b) What is the relationship between l 1 and l 2? 6) Given a point ( 3, 6) and a line y = 2x 8: a) What is the slope of the line? b) What is the slope of any line parallel to the given line? c) Write an equation of a line through the point and parallel to the line. d) What is the slope of any line perpendicular to the given line? Explain. 7) Given the point (0, 7) and the line y = 1 x a) What is the slope of any line parallel to the given line? Explain your answer. b) Write an equation of a line through the point and perpendicular to the line. c) If a line is perpendicular to y = 1 x + 5, will it be perpendicular to x 2y = 14? Explain. 2

20 8) Find an equation of a line through ( 3, 1 ) and parallel to the line: 2 a) x = 9 b) y = 7 c) What can you conclude about your answer in parts (a) and (b)? A line segment with one endpoint on a line and perpendicular to the line is called a normal segment to the line. A line containing the normal segment is called the normal line. 9) Given AB with coordinates A(5, 7) and B(8, 2). a) Find an equation for the normal line through A. b) Find an equation for the normal line through B. 10) Given the line 3x 6y = 15, find the equation of the normal segment passing through the point (8, -4).

b) By rotating the segment 90, what type of lines are we creating? c) What is the slope of the rotated segment?

21 ROTATING 90 ABOUT A POINT [7] The image of a point rotated 90 (clockwise or counterclockwise) about any point, lies on the line to the original segment. 1) Plot the endpoints of segment AB: A(3, 7) to B(10, 1). a) What is the slope of the original segment? b) By rotating the segment 90, what type of lines are we creating? c) What is the slope of the rotated segment? d) Find the image of point B after a rotation of 90 about point A. e) Find the image of point B after a rotation of -90 about point A. 2. Draw segment DE with endpoints D(1, 6) and E(-3, 5). Rotate the segment -90 about point E.

22 3. A triangle with endpoints A(2, 1), B(0, 4), and C(-5, 6) is rotated 90 about the point (0, 4). Find the coordinates of the image. 4. Use the grid at the right showing points O(0,0), P(3, 1), and Q(2,3) on the coordinate plane. Determine whether OP and OQ are perpendicular. 5. Plot the following points on a coordinate plane: O(0,0), A(6,4), and B( 2,3). a) Draw ABO. b) Is ABO a right triangle? Justify your answer using slopes.

23 6. Carlos thinks that the segment having endpoint A(0, 0) and B (6,0) is perpendicular to the segment with endpoints A(0,0) and C( 2, 0). Do you agree? Why or why not? 7. Given A(0,0) and B(3, 2), find possible coordinates of a point C so that AC AB.

24 Writing the Equations of Special Segments in a Triangle [8] The median of a triangle is a segment drawn from a vertex to the of the opposite side. 1. Triangle PQR has coordinates A(2, -6), B(-4, 6) and C(8, 6). a) Write an equation of the line containing the median drawn from vertex A. b) Write an equation of the line containing the median drawn from vertex B. c) Write an equation of the line containing the median drawn from vertex C. d) What are the coordinates of the centroid of the triangle?

25 A perpendicular bisector is a segment which is to a segment at its. 2. Triangle PQR has coordinates D(3, 5), E(3, -1) and F(-5, -1). a) Write an equation of the perpendicular bisector of DE. b) Write an equation of the perpendicular bisector of EF. c) Write an equation of the perpendicular bisector of DF. d) What are the coordinates of the circumcenter of the triangle?

26 Recall: An altitude of a triangle is a segment drawn from a vertex of the triangle to the line containing the opposite side. An altitude is a normal segment. 4. Triangle PQR has coordinates P(-6, 4), Q(4, 4) and R(-3, -3). a) Write an equation of the line containing the altitude drawn from vertex P. b) Write an equation of the line containing the altitude drawn from vertex Q. c) Write an equation of the line containing the altitude drawn from vertex R. d) What are the coordinates of the orthocenter of the triangle?

27 Dividing Segments Proportionally [9] a. Draw right triangle ABC. b. What is the length of AC? c. What is the length of BC? d. Mark the halfway point on AC and label it P. What are the coordinates of P? e. Mark the halfway point on BC and label it R. What are the coordinates of R? f. Draw a segment from P to AB perpendicular to AC. Mark the intersection M. What are the coordinates of M? g. Draw a segment from R to AB perpendicular to BC. What do you notice? h. Describe how you found point M. Midpoint Formula The midpoint formula can be used to find the midpoint of a segment. 1. Find the coordinates of M, the midpoint of ST, for S(-6, 3) and T(1, 0). Finding the midpoint of a segment divides the segment into equal parts. The segments have a ratio of : 2. Find the coordinates of J if K(-1, 2) is the midpoint of JL and L has coordinates (3, -5).

28 3. a. Find the point, C, that partitions the segment, AB, with endpoints A(-3, 1) and B(5, 3), into a ratio of 1:3. b. What if we wanted to find the point that is 1 4 of the way along segment AB, closer to A than B? How is this the same/different as part a? 4. Find the point on the directed segment from (-4,5) to (12,13) that divides it into a ratio of 1:7. 5a. Given points A(-4,5) and B(12,13), find the coordinates of the point C, that sits 2 5 of the way along segment AB, closer to A than it is to B. b. Given points A(-4,5) and B(12,13), find the coordinates of the point D, that sits 2 5 of the way along segment AB, closer to B than it is to A.

29 6. Given points P(10,10) and Q(0,4), find point R on PQ such that PR 7. RQ 3 7. Given points A(3,-5) and B(19,-1), find the coordinates of point C that sits 3 8 of the way along AB, closer to A than to B.

Module Four: Connecting Algebra and Geometry Through Coordinates

NAME: Period: Module Four: Connecting Algebra and Geometry Through Coordinates Topic A: Rectangular and Triangular Regions Defined by Inequalities Lesson 1: Searching a Region in the Plane Lesson 2: Finding