EQUATIONS OF ALTITUDES, MEDIANS, and PERPENDICULAR BISECTORS Steps to Find the Median of a Triangle: -Find the midpoint of a segment using the midpoint formula. -Use the vertex and midpoint to find the slope of the median. -Use the vertex or midpoint to help find the y-intercept of the line. -Write the equation for the median. Example: In triangle PQR with vertices P(-5, 3) Q(3, 7) and R(1, 3), find the equation of median RM. Steps to Find the Altitude of a Triangle: -Find the slope of the segment that the altitude will intersect. -Calculate the negative reciprocal of this slope. This will be the slope of the altitude. -Use the vertex and the slope to help find the y-intercept. -Write the equation for the altitude.
Example: In triangle EFG with vertices E( 4, 5) F(2, 10) and G( 15, 3), Find the equation of altitude GH. Steps to Find the Perpendicular Bisector of a Segment: -Find the midpoint using the midpoint formula. -Find the slope of the line segment. -Calculate the negative reciprocal of this slope. This will be the slope of the perpendicular line. -Use the midpoint and the slope to help find the y-intercept. -Write the equation for the perpendicular bisector. Example: In triangle ABC with vertices A(1, 3) B( 3, 5) and C(6, 8). Find the equation of the perpendicular bisector of side AB.
TRIANGLES AND QUADRILATERALS IN THE COORDINATE PLANE Example: Given ELF with vertices E( 2, 2), L(4, 0) and F(6, 2) (a) Find point A, the midpoint of EF (b) Find point B, the midpoint of LF. (c) Show that AB EL (d) Show that AB = ½ EL
Example: If the coordinates of the vertices of quadrilateral CARD are C( 4, 1), A( 2, 3), R(1, 0) and D( 1, 2), show that CARD is a rectangle. (a) Show it is a parallelogram by showing that the diagonals bisect each other *Find the midpoint of each diagonal and show they are the same. (b) Show the diagonals are congruent OR Show the sides form right angles *Find their distance and show they *Find slopes and show opposite are the same reciprocal Example: If the vertices of quadrilateral PEAR are P( 3, 0), E(0, 4), A(5, 6) and R( 1, 4) (a) Find the slopes of all the sides to show that it is a trapezoid.
PROOFS WITH VARIABLES Example: Given ELF with vertices E(0, 0), L(4a, 0) and F(4a, 4c) (a) Find point A, the midpoint of EF, and point B, the midpoint of LF. (b) Show that AB EF. Example: The vertices of quadrilateral MATH are M(0, 0), A(b, c), T(a+b, c) and H(a, 0) (a) Using coordinate geometry, show that the diagonals bisect each other. (b) Using coordinate geometry, show that MH AT
Example: If the coordinates of the vertices of quadrilateral ABCD are A(0, 0), B(b, y), C(a+b, y) and D(a, 0) (a) Prove that ABCD is a parallelogram. (b) Prove that ABCD is not a rhombus. Example: The diagram shows ABCD, where a b, a 0, b 0. Prove ABCD is an isosceles trapezoid.
CIRCLE EQUATIONS Circle the set of all points equidistant from a fixed point. Recall the Distance Formula d = (x 2 x 1 ) 2 + (y 2 y 1 ) 2 Equation of a Circle A circle with radius (r) and center (h, k) has the equation: r = (x h) 2 + (y k) 2 which equals Example: Write the equation of a circle whose center has coordinates (5, 3) and radius has length of 6. Example: Which point lies on the circle x 2 + y 2 = 49? (a.) (0, 0) (b.) (3, 4) (c.) ( 4, 3) (d.) (0, 7) Example: The center of a circle is at the origin and passes through the point (2, 4) (a) Write the equation of the circle. (b) Find two other points on the circle. Example: The equation of a circle is (x 8) 2 + (y + 11) 2 = 25
(a) Find the length of the radius and the coordinate of the center. (b) Find two points on the circle. Example: The center of a circle is at ( 1, 2) and the circle passes through the point (4, 3) (a) Find the length of the radius of the circle. (b) Write the equation of the circle. Example: Write the equation of a circle if the coordinates of the endpoints of the diameter are (2, 1) and (4, 5).
GRAPHING EQUATIONS A linear and quadratic system of equation can be solved graphically. You will find that there are either two solutions, one solution or no solutions. This is illustrated below. The graph of y = ax 2 + bx + c has the line x = b 2a as its axis of symmetry. The vertex has coordinates ( b y = x 2 +4x +3 x x 2 + 4x + 3 y 2a, f ( b 2a )) Types of Graphs LINES CIRCLES PARABOLAS y = mx + b (x h) 2 + (y k) 2 = r 2 y = ax 2 + bx + c
Example: How many solutions are there to the following system of equations? y = 3 y = x 2 4x 1 Example: How many solutions are there to the following system of equations? y = (x 3) 2 2y + x = 1 Example: How many solutions are there to the following system of equations? 2x 3y = 2 x 2 + y 2 = 4