Courtney, Sam, Anastasia, Jeremy, Alan and Matthew Unit 2: Analytic Geometry Equation of a line­ y= mx +b Distance­ 2 2 2 Equation of a Circle­ x +​ y =​ r ​ ​ ​ Key Terms and Definitions 1. Median­ A line that joins the vertex of a triangle to the midpoint of the opposite side. ​ Perpendicular Bisector­ Perpendicular line segment that passes through the midpoint ​ Circumcircle­ A circle that passes through all vertices. ​ Centroid­ The point where three medians of a triangle intersect. ​ Circumcentre­ The point where three perpendicular bisectors meet. ​ Angle Bisector­ Line segment that splits the angle into two equal parts. ​ Altitude­ A line segment from the vertex perpendicular to the opposite side. ​ Incentre­ The point where the angle bisectors meet ​ Orthocentre­ The point where three altitudes of a triangle intersect. ​ Midsegment­ A line segment formed by joining the midpoints of two adjacent sides ​ Incircle­ The largest circle that can be inscribed in a triangle ​ Equation of Circle with Centre (0,0) x and y are the points on the circle. Sub in the points and calculate the radius. Points on, inside or outside the circle, 4 quick points To find 4 quick points: Find the radius of the circle, then add to x and y. Left side right side check using the above equation Left side: Right Side: Sub in the values If LS>RS then Point is outside the circle If LS=RS then Point is on the circle If LS<RS then Point is in the circle Length Of Altitude (Alan) ​ In order to find the length of the altitude of a triangle by using coordinates, we must find the POI between line BC, and AD. 1 ­ Draw altitude and plot co­ordinate on the perpendicular intersection (altitude line through the other line and label it (e.x. D). 2 ­ Find the equation of line segment (BC) by using your slope formula and substituting the point (B or C) in y = mx + b. 3 ­ Now you must determine the equation of line AD by using the negative reciprocal slope of the bottom line you just found (BC) and substituting the the coordinates of A in order to find 1 the slope equation. (­ m of Line AD = mperpendicular of Line BC). 4 ­ After you have found both equations use substitution to solve to find the POI (mx +b = mx + b). 5 ­ Now that you know the coordinates of D, you use the distance formula to find the length of AD. (d= ­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­ Example: A triangle is formed on the coordinates A(­1,5), B(­3,­2), and C(3,4). What is the length of the altitude? (See solution below) Solution: y2 − y1 mBC = x2 − x1 4 − (−2) = 3 − (−3) 6 = 6 = 1 y = 1x + b Substitute C into equation 4 = 1(3) + b 1 = b (1) y = x + 1 mAD = ­1 ⊥ y = ­x + b Substitute A into equation 5 = ­(­1) + b (2) y = ­x + 4 Sub (1) into (2) x + 1 = ­x + 4 x + x = 4 ­ 1 2x = 3 3 x = 2 Sub x into (1) 3 y = 2 + 1 5 y = 2 d = √(x2 − x1)2 + (y2 − y1)2 d = (3 + 1)2 + (5 − 5)2 √ 2 2 25 d = √ 2 25 Therefore the length of the altitude is √ 2 . Constructing a Median: ­ Measure line segment and plot the midpoint ­ Draw a line from the opposite vertex to the midpoint Constructing Altitudes: ­ Measure a 90 degree angle with the vertex’s opposite side (use right angle tool) ­ Draw the line Constructing Perpendicular Bisectors: ­ Find the midpoint ­ Draw a line perpendicular through the midpoint Constructing the Circumcircle: ­ Use the circumcentre as the center of the circle and it will touch all 3 vertices Constructing Angle Bisectors: ­ Measure the angle at the vertex ­ Draw a line through half of that angle Constructing the Incircle: ­ Using the incentre as the center of the circle, it will touch one point on each side of the inside of the triangle Finding Equation of the Median: 1. Find midpoint of BC (D) 2. Find slope of median (AD) 3. Sub midpoint (D) or (A) into y=mx+b A (­1,4) B (7,2) C (1,­6) MAB = (7+1/2, 2­6/2) ​ ​ D = (4, ­2) m = ­2­4/4+1 =­6/5 y = ­6/5 x +b ­2= ­6/5(4) +b ­2=­24/5+b 14/5=b y= ­6/5x+14/5 Finding Equation of the Altitude: 1. Find slope of BC 2. Find perpendicular slope (slope of altitude) 3. Sub A into y=mx+b (opposite point) A (­1,4) B (7,2) C (1,­6) m= ­6­2/1­7 m=4/3 m perp = ­¾ sub in A y = ­3/4x +b 4= ­¾(­1) +b 13/4=b y= ­3/4x+13/4 Finding Equation of the Perpendicular Bisector: 1. Find midpoint of AB (D) 2. Find slope of AB 3. Find perpendicular slope (perpendicular bisector’s slope) 4. Sub. midpoint AB into y=mx=b A (­1,4) B (7,2) C (1,­6) MAB = (7+1/2, 2­6/2) ​ ​ D = (4, ­2) y=4x+b sub D mAB=2­4/7+1 ­2=4(4)+b ​ ​ =­¼ 14=b mperp=4 y=4x+14 ​ ​ Practice Questions: 1. Find the equation of the median from A to BC fro a triangle with the coordinates ​ ​ of: A(­7,4), B(­2,­9), C(1,10) 2. Find the equation of the altitude from A to BC for a triangle with the coordinates ​ ​ of: A(5,­4), B(­2,6), C(0,­8) 3. Find the equation of the perpendicular bisector for a line segment with the ​ ​ coordinates of: A(5,­4), B(­7,6) Solutions: ​ 2. y= 7/10x­87/10 3. y=6/5x+11/5 Types of Quadrilaterals Quadrilateral­ Any four sided closed shape o When you are given four vertices and need to find out what type of quadrilateral it ​ ​ is there are two steps you need to take 1) Find the Slopes ​ ​ 2) Find the Lengths (distances) ​ ​ Rhombus or Square o You can narrow it down to these two shapes if all your distances are the same ​ ​ o If the slopes are perpendicular then it is a square, if not it is a rhombus ​ ​ Parallelogram or Rectangle o You will have two pairs of equal side distances ​ ​ o For the slopes it is the same, if they are perpendicular it is a rectangle, if not it is a ​ ​ parallelogram Kite o Your distances will have two equal pairs ​ ​ o All your slopes will be different ​ ​ Trapezoid o Your distances will be different ​ ​ o You will have two of the same slopes which will tell you they are parallel which ​ ​ tells you it should be a trapezoid If you connect all midsegments of a quadrilateral a parallelogram will be formed Example­ O(0,0) P(3,5) Q(8,6) R(5,1) What type of quadrilateral is it? Answer­ Parallelogram Classifying triangles­ Find the side length and then find the slopes Scalene­ all different lengths Isosceles­ two sides equal Equilateral­ all sides equal Right­ 90 degree angle .