ࡱ> '` bjbjLULU LG.?.?E4 ;;;8;<lZ= z?z?z?z?kHkHkH$th܊>kH H^kHkHkHz?z?†3K3K3KkH(z?z?3KkH3K3KNoz?N= g\;H0+\؆0GLcIT`kHkH3KkHkHkHkHkHJ|kHkHkHkHkHkHkHz(Dz((4\"~ NOTE: A great way to study is to turn to the examples given in the text and try to solve them without looking at how it is done in the text. Once you have finished working through the problem by yourself, look through the solution outlined in your text book! Chapter 2 Review Analytic Geometry 2.1 Midpoint of a Line Segment (median, parallel & perpendicular lines, right bisectors) CONCEPTSEXAMPLE PROBLEMSTo find the midpoint, M, of a line segment joining  EMBED Equation.3 and  EMBED Equation.3 , use the midpoint formula,  EMBED Equation.3 . Parallel Lines The slopes of 2 parallel lines are the same. The y-intercepts may be different or the same. Perpendicular Lines The slopes of 2 perpendicular lines are negative reciprocals. This means that when they are multiplied together, the answer is -1. Bisector A bisector is a line which intersects another line at its midpoint. Right Bisector A right bisector is a line which intersects another line at its midpoint at a 90 angle. For the line segment ST, S has coordinates (6, 2) and T is on the y-axis. The midpoint, M, of ST is on the x-axis. Find the coordinates of T and M. Determine the equation of the right bisector of a line with end points at (-2, 5) and (-4, -3). Determine the equation of the median from vertex A to midpoint BC of  EMBED Equation.3  with vertices at A(-1, 8), B(-4, 2), & C(0, -2).  EMBED Equation.3 has vertices A(3, 4), B(-5, 2), and C(1, -4). Determine an equation for: CD, the median from C to AB GH, the right bisector of AC To find the equation of a right bisector:  EMBED Equation.3 has vertices A(3, 4), B(-5, 2), and C(1, -4). Determine the equation of GH, the right bisector of AC  1. Find the midpoint, G, of AC:  EMBED Equation.3  2. Find the slope of AC: A(3, 4) & C(1, -4)  EMBED Equation.3  3. Find the equation of right bisector: G(2,0) mGH= - (neg. rec. of mAC)  EMBED Equation.3  OR  EMBED Equation.3  so, y = -x +  To find the equation of a median of a triangle:  EMBED Equation.3 has vertices A(3, 4), B(-5, 2), and C(1, -4). Determine an equation for CD, the median from C to AB  1. Find midpoint, D, of side AB:  EMBED Equation.3  2. Find slope of median using points A & D:  EMBED Equation.3   3. Find equation of median using point-slope form or slope - y-intercept form: CDm =  EMBED Equation.3  C(1, -4)  EMBED Equation.3  OR  EMBED Equation.3 See examples on pages 60(find midpoint), 62 (eq. for median of triangle), 63(Eq. for right bisector). Key concepts on page 65.  2 2.2 Length of a Line Segment CONCEPTSEXAMPLE PROBLEMSTo find the length of a line segment joining  EMBED Equation.3 and  EMBED Equation.3 , use the formula  EMBED Equation.3 .Determine the perimeter of the quadrilateral with vertices A(3,2), B(-2, 4), C(-4, -2), and D(4, -1) (a) as an exact solution (b) as an approximate solution to the nearest tenth.Key Concepts page 76.See examples on pages 74-76. 2.3 Apply Slope, Midpoint, and Length Formulas (Shortest distance from a point to a line) CONCEPTSEXAMPLE PROBLEMSDetermine equations for lines. Determine when an angle is a right angle. Classify triangles by side length (equilateral, isosceles, scalene). Determine type of quadrilateral by slopes & length of lines (parallelogram, rectangle).The equation of a circle with centre  EMBED Equation.3  is  EMBED Equation.3 . The points M(1,4) and N(4, -1) are endpoints of chord MN. Find the distance from the centre of the circle to the chord, to the nearest tenth.  EMBED Equation.3 has vertices A(3, 4), B(-5, 2), and C(1, -4). Determine an equation for AE, the altitude from A to BC.  To determine the equation of an altitude: Determine the equation of AE, the altitude from A to BC 1. Find the slope of BC: B(-5, 2) & C(1, -4)  EMBED Equation.3   2. Find equation of altitude using point-slope form: A(3, 4) and negative reciprocal of mBC  EMBED Equation.3  OR Use y=mx+b:  EMBED Equation.3  so, y = x + 1To determine the shortest distance from a point to a line: To determine the distance from a point, P, to a line, AB, when given the coordinates of P, and the equation of AB: Determine the slope of AB in order to find the slope of a perpendicular line. Negative reciprocal. Determine the equation of the perpendicular line to AB that passes through your point P using the point-slope form of a line.  EMBED Equation.3  You now have two equations for lines that intersect. Solve this linear system using substitution or elimination. equation of line AB which was given equation just found FINALLY find the length of the line segment between point P, and the point of intersection which you just found.  EMBED Equation.3 Key Concepts page 88.See examples on pages 82-87. 2.4 Equation for a Circle CONCEPTSEXAMPLE PROBLEMSAn equation of the circle with centre  EMBED Equation.3  and radius  EMBED Equation.3 is  EMBED Equation.3 . (More circles in section 3.5)Determine the radius of a circle given its centre (0, 0) if a point on its circumference is (2, 7). Round the radius to the nearest tenth, if necessary. Write an equation for the circle.Key Concepts page 96.See examples on pages 94, 95. See Also Review on pages 100 to 103. Chapter Test on pages 104 and 105.     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