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EQUATIONS OF ALTITUDES, MEDIANS, and BISECTORS

Steps to Find the of a : -Find the of a segment using the midpoint formula. -Use the and midpoint to find the of the median. -Use the vertex or midpoint to help find the y-intercept of the . -Write the 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 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 . -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 IN THE COORDINATE

Example: Given ∆ELF with vertices E(–2, –2), L(4, 0) and F(6, –2)

(a) Find 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 CARD are C(–4, 1), A(–2, 3), R(1, 0) and D(–1, –2), show that CARD is a .

(a) Show it is a by showing that the bisect each other *Find the midpoint of each and show they are the same.

(b) Show the diagonals are congruent OR Show the sides form right *Find their distance and show they *Find 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 .

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 .

Example: The diagram shows ABCD, where a≠b, a≠0, b≠0. Prove ABCD is an isosceles trapezoid.

CIRCLE

Circle – the set of all points from a fixed point.

2 2 Recall the Distance Formula 푑 = √(푥2 − 푥1) + (푦2 − 푦1)

Equation of a Circle – A circle with radius (r) and center (h, k) has the equation:

푟 = √(푥 − ℎ)2 + (푦 − 푘)2 which equals ______

Example: Write the equation of a circle whose center has coordinates (5, –3) and radius has of 6.

Example: Which point lies on the circle x2 + y2 = 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 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.

−푏 The graph of y = ax2 + bx + c has the line x = as its axis of . 2푎

−푏 −푏 The vertex has coordinates ( , 푓 ( )) 2푎 2푎 y = x2 +4x +3 x x2 + 4x + 3 y

Types of Graphs

LINES y = mx + b (x – h)2 + (y – k)2 = r2 y = ax2 + bx + c

Example: How many solutions are there to the following system of equations?

y = 3 y = x2 – 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 x2 + y2 = 4