Midpoint of a Line Segment Pg. 78 # 1, 3, 4-6, 8, 18. Classifying Figures on a Cartesian Plane Quiz ( )

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1 UNIT 2 ANALYTIC GEOMETRY Date Lesson TOPIC Homework Feb. 22 Feb. 23 Feb. 24 Feb. 27 Feb Mar Mar Mar Mar Mar. 7/ Mar Midpoint of a Line Segment Pg. 78 # 1, 3, 4-6, 8, 18 Length of a Line Segment In Class Assignment - Problems Pg. 86 # 1, 3, 5ii, 6, 7, 9, 17 Equation of a Circle Pg. 91 # (1 6)doso, 7i, iii, 8, 10, 13, 14, 18 Equations of Altitudes/Bisectors and Medians Equations of Altitudes/Bisectors and Medians Classifying Figures on a Cartesian Plane Quiz ( ) WS 2.4 # 1ac, 2, 3bd, 5, 6aceg Pg. 79 # 7, 13ac WS 2.5 # 1ac, 2, 3bd; # 1ace, 4; # 1ace, 2a Pg. 101 # 1 3, 7, 8, 9a, 12, 17 Verifying Properties of Geometric Figures Pg. 109 # 2, 3, 5, 8, 14 Exploring Properties of Geometric Figures - Centroid/Circumcentre/Orthocentre Using Coordinates to Solve Problems Quiz ( ) Pg. 120 # 9 12, 14 Pg. 120 # 16, 20, 21 Review for Unit 2 Test Pg. 124 # 1 3, 5, 7 11, 13, 15, 17, 18, 19a, 21 23, 25 TEST- UNIT 2

2 MPM 2D GEOMETRY DEFINITIONS TRIANGLES Equilateral a triangle with 3 sides of equal length and 3 angles of equal measure. Isosceles a triangle with 2 sides of equal length. The angle opposite the equal sides are also of equal measure. Scalene a triangle with no equal sides or angles. Right a triangle with one 90 angle. QUADRILATERALS Parallelogram a quadrilateral where opposite sides have are parallel. Rectangle a parallelogram in which all interior angles are 90. Rhombus a parallelogram with all sides of equal length. Square A rectangle with 4 sides of equal length MEDIAN a line that joins the vertex of a triangle to the midpoint of the opposite side. FT is one median of DEF F E D T PERPENDICULAR BISECTOR A line that is perpendicular to another line segment and passes through the midpoint of the line segment. ie: In ABC, the line segment from M is the perpendicular bisector of CB. A C M B

3 ALTITUDE the line segment representing the height of a polygon, drawn from a vertex perpendicular to the opposite side. P PS is the altitude of PQR. R S Q MIDSEGMENT the line segment formed by joining the midpoints of 2 adjacent sides of a polygon Midsegment CENTROID the centroid of a triangle can be found by finding the equations of 2 of the median lines,then finding the point of intersection of those two lines. It can also be found by calculating the average of the x- and y-coordinates of all three vertices of the triangle. Diagram Pg CIRCUMCENTRE can be found by finding the equations of the perpendicular bisectors of 2 sides of a triangle, then finding the point of intersection of those two lines. Diagram Pg ORTHOCENTRE can be determined by finding the equations of 2 of the altitude lines, then finding the point of intersection of those two lines. Diagram Pg PERPENDICULAR LINES have slopes that are the negative reciprocal of each other. m l 1 1 l m 2 PARALLEL LINES have equal slopes. ml1 ml 2 SLOPE Y-INTERCEPT FORM OF EQUATION OF A LINE y mx b m y x 2 2 y x 1 1 STANDARD FORM OF EQUATION OF A LINE Ax by C 0, where A, B, C R

4 MPM 2D Lesson 2.1 Midpoint of a Line Segment Find the midpoint of the line segment below. y x The coordinates of the midpoint of a line segment are the averages of the endpoints of the line segment. A x, ) ( 1 y1 M x 1 x 2 2, y 1 y 2 2 B x, ) ( 2 y 2 to find the midpoint given the two endpoints, where x, y ) is the midpoint. ( M M x M x 1 x 2 and 2 y M y 1 y 2, 2 Ex. 1 Find the coordinates of the midpoint of the line segment with endpoints A( 9, 3) and B(5, 7).

5 Ex. 2 One end of a line segment is P( 6, 3) and the midpoint is M(4, 1). Find the other endpoint Q. Ex. 3 A circle has a radius with endpoints E(2, 4) and F(-4, -8). Find two possible endpoints for the diameter that contains the radius. There are 2 possible circles. B E E F F A I. Let B be the other endpoint of the diameter. II. Let A be the other endpoint of the diameter. Pg. 78 # 1, 3, 4-6, 8, 18

6 MPM 2D Lesson 2.2 Length of a Line Segment To find the distance between two points, we can use the Pythagorean Theorem. y x Ex. 1 Find the length of the line segment connect the two points A and B below. y B x A 5 6 Ex. 2 Find the distance between P(-3, 7) and (7, -8).

7 Ex. 3 A plane is flying from Toronto to Halifax. When at coordinates (125, 309) the plane develops engine trouble. Ottawa International Airport is at coordinates (97, 215) and Pierre Elliot Trudeau Airport is at coordinates (139, 412). To which airport should the plane divert? Pg. 86 # 1, 3, 5ii, 6, 7, 9, 17

8 MPM 2D Lesson 2.3 EQUATION of a CIRCLE with CENTRE (h, k) CIRCLE the set of all points in a plane that are the same distance from a fixed point, the centre. The distance from the centre of a circle to any point on the circle is called the radius. If the centre of the circle is at the point (h, k) and the radius is r units, then the equation (x h) 2 + (y k) 2 = r 2 is the equation of the circle. (h, k). Ex. 1 Write the equation of the circle with centre (0, 0) and radius of 2 1 units. Ex. 2 A circle with centre (0, 0) passes through the point (6, -8). a) Find the equation of the circle. b) What is the other endpoint of the diameter that passes thru (6, -8)?

9 Ex. 3 A stone is dropped in the water and sends out ripples whose radius increases at 5 cm/s. Find the equation of the outer ripple 12 s after the stone is dropped. Ex. 4 Find the equations of the following circles. a) centre (-3, 2) with a radius of 4 b) centre ( 2, -5) passing through the point ( 9, 10) Ex. 5 Given: (x 1) 2 + (y + 3) 2 = 50 Determine if the following points are inside, on, or outside the circle. a) (-2, 7) b) (4, 1) c) (6, 2)

10 Ex. 4 A truck with a wide load is approaching a tunnel that is shaped like a semicircle. The maximum height of the tunnel is 5.25 m high. The load is 8 m wide and 3.5 m high. The driver uses his grade 10 math skills to determine if the load will it fit through the tunnel. Must the driver take another route? Pg. 91 # (1 6)doso, 7i, iii, 8, 10, 13, 14, 18

MPM 2D Lesson 2.4 Equations of Altitudes, Bisectors and Medians To find the equation of a line you need: the slope of the line a point on the line OR two points on the line Ex.

11 MPM 2D Lesson 2.4 Equations of Altitudes, Bisectors and Medians To find the equation of a line you need: the slope of the line a point on the line OR two points on the line Ex. 1 Find the equation of the line through A(2, 3) and B(3, 5). General Form of Equation of a Line: y mx b Standard Form of Equation of a Line: Ax By C 0, A, B, C I, A 0 Ex. 2 Find the equation of the line through P(5, 4) and perpendicular to AB, for A(-1, 3) and B(6, -4).

12 Ex. 3 Find the equation of the line through Q(-1, 0) and parallel to 2x 3y 1 0. Ex. 4 ABCD has vertices A(3, 7), B(-4, -2), C(6, 1) and D(-1, 0). Find the equation of the midsegment of sides AB and BC. WS 2.4 # 1ac, 2, 3bd, 5, 6aceg

13 MPM 2G Lesson 2.5 Equations of Altitudes, Bisectors and Medians MEDIAN a line that joins the vertex of a triangle to the midpoint of the opposite side. FT is one median of DEF F E T D Ex. 1 If ABC has vertices A(12, 4), B(-6, 2), and C(-4, -2), find the equation of the median from B.

14 PERPENDICULAR BISECTOR A line that is perpendicular to another line segment and passes through the midpoint of the line segment. ie: In ABC, the line segment from M is the perpendicular bisector of CB. A C M B Ex. 2 ABC has vertices A(12, 4), B(-6, 2), and C(-4, -2), find the equation of the perpendicular bisector of AB. ALTITUDE the line segment representing the height of a polygon, drawn from a vertex perpendicular to the opposite side. P PS is the altitude of PQR. R S Q Ex. 3 ABC has vertices A(-1, 4), B(-1, -2), and C(5, 1), find the equation of the altitude to A. Pg. 79 # 7, 13ac WS 2.5 # 1ac, 2, 3bd; # 1ace, 4; # 1ace, 2a

MPM 2D Lesson 2.6 Classifying Figures on a Cartesian Plane SHAPE DEFINITION SKETCH Scalene Triangle A 3-sided polygon where there are no sides of equal length.

Find lengths of all sides. Right Triangle Parallelogram Rectangle Square Rhombus A 3-sided polygon which has one 90 angle. A quadrilateral with opposite sides that are parallel.

-find the slopes of all sides -show that 2 slopes are negative reciprocals of one another -find the slopes of all sides -show that opposite sides have the same slope -find the length of all sides

15 MPM 2D Lesson 2.6 Classifying Figures on a Cartesian Plane SHAPE DEFINITION SKETCH Scalene Triangle A 3-sided polygon where there are no sides of equal length. What is required in order to classify the shape Find lengths of all sides. Isosceles Triangle A 3-sided polygon which has 2 sides of equal length. Find lengths of all sides. Equilateral Triangle A 3-sided polygon which has 3 sides of equal length. Find lengths of all sides. Right Triangle Parallelogram Rectangle Square Rhombus A 3-sided polygon which has one 90 angle. A quadrilateral with opposite sides that are parallel. A parallelogram in which all the angles are 90. A rectangle in which all sides are of equal length. A parallelogram in which all sides are of equal length. -find the slopes of all sides -show that 2 slopes are negative reciprocals of one another -find the slopes of all sides -show that opposite sides have the same slope -find the length of all sides -show opposite sides have the same length -show that adjacent sides have slopes are negative reciprocals -find lengths of all sides -find slopes of sides -only need to prove one 90 angle -find the length of all sides -if all sides are equal, it is a rhombus - a square is a special rhombus

16 Ex. 1 Show that quadrilateral EFGH with vertices E ( 5, 4), F ( 2, 8), G (6, 2), H (3, 2) is a rectangle. Ex. 2 Given: A ( 1, 3), B (1, 7), and C (5, 5) a) verify that ABC is a right triangle. b) determine whether or not ABC is isosceles.

17 Ex. 3 Determine the type of quadrilateral defined by the following vertices. A(2, 3), B(5, -1), C(10, -1), D(7, 3) Pg. 101 # 1 3, 7, 8, 9a, 12, 17

18 MPM 2D Lesson 2.7 Verifying Geometric Properties Ex. 1 Show that the midsegments of quadrilateral PQRS with vertices P(-2, -2), Q(0, 4), R(6, 3) and S(8, -1) form a parallelogram. y Q R x 1 P S

19 Ex. 2 a) Show that A(-4, 3) and B(3, -4) lie on the circle x 2 + y 2 = 25. b) Show that the perpendicular bisector of chord AB passes through the centre of the circle. a chord is the line segment joining two points on a curve. The term is often used to describe a line segment whose ends lie on a circle. Pg. 109 # 2, 3, 5, 8, 14

20 MPM 2D Lesson 2.8 Exploring Properties of Geometric Figures - Centroids/Circumcentres/Orthocentres CENTROID the centroid of a triangle can be found by finding the equations of 2 of the median lines,then finding the point of intersection of those two lines. It can also be found by calculating the average of the x- and y-coordinates of all three vertices of the triangle. It is the centre of mass of the triangle. Diagram Pg CIRCUMCENTRE can be found by finding the equations of the perpendicular bisectors of 2 sides of a triangle, then finding the point of intersection of those two lines. It is the point which is equidistant from each vertices. Diagram Pg ORTHOCENTRE can be determined by finding the equations of 2 of the altitude lines, then finding the point of intersection of those two lines. Diagram Pg Ex. 1 For the triangle with vertices A(4, 9), B(1, 1), and C(8, -4). a) Find the centroid.

21 b) Find the circumcentre.

22 c) Find the orthocentre. Pg. 120 # 9 12, 14

23 MPM 2D Lesson 2.9 Using Coordinates to Solve Problems Ex. 1 The closest power line to a parking lot runs along a line through the points A(0, 4) and B(12, 10). At what point should contractors connect a power line so that they use the least amount of cable to reach a lamppost at coordinates L(6, 19)? How much cable, correct to one decimal place, will they use? Pg. 120 # 16, 20, 21

Chapter 2 Diagnostic Test

Chapter 2 Diagnostic Test Chapter Diagnostic Test STUDENT BOOK PAGES 68 7. Calculate each unknown side length. Round to two decimal places, if necessary. a) b). Solve each equation. Round to one decimal place, if necessary. a)