Analytic Geometry
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 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