P.2: Absolute Value, Distance Formula, Midpoint Formula, Equations of Circles

First, recall the Cartesian Plane: (also called the rectangular coordinate system) First Quadrant y Ordered pair: y-axis x-coordinate P(3, 1) Second Quadrant P(3, 1) y-coordinate x origin x-axis Fourth Quadrant Third Quadrant

Absolute Value a, if a > 0 Notation! |a|= –a, if a < 0 …tells magnitude of a number… The absolute value of a real number a is: a, if a > 0 Notation! |a|= –a, if a < 0 0, if a = 0 Ex: |5| = 5 Ex: | – 6| = 6 – Ex: |–3| = 3

Distance Formula |a – b| d = (x – x ) + (y – y ) Let a and b be real numbers. The distance between a and b is: |a – b| The distance between points P(x , y ) and Q(x , y ) in the coordinate plane is: 1 1 2 2 2 2 d = (x – x ) + (y – y ) 1 2 1 2 (derived via the Pythagorean Theorem)

Midpoint Formula , a + b 2 a + c b + d 2 2 The midpoint of the line segment with endpoints a and b is: a + b (Number Line) 2 The midpoint of the line segment with endpoints (x, b) and (c, d) is: a + c , b + d 2 2 (Coordinate Plane)

Guided Practice d= 9 d = 2 29 Find the distance between the points: –2, –11 (–3, 1), (7, –3) d= 9 d = 2 29

Guided Practice Midpt = – 4.05 Midpt = (–½, 5) Find the midpoint of the line segment with the given endpoints: 2.3, –10.4 (5, 8), (–6, 2) Midpt = – 4.05 Midpt = (–½, 5)

Standard form equation for a circle What is a circle??? The set of points in a plane at a fixed distance (radius) from a fixed point (center) y (x, y) Let’s use the distance formula to find the radius: r (h, k) x Then, what happens if we square both sides… Standard form equation for a circle

Guided Practice: Find the standard form equation of the circle. Center (–3, 6), radius 9 Center (0, 2), radius

This expression means that the distance from x Guided Practice: Describe the set of real numbers that satisfy: This expression means that the distance from x to 2 must be less than 3… Thus, x must be between –1 and 5:

With only two sides congruent, the Guided Practice: Prove that the triangle determined by the points (3,0), (–1,2), and (5,4) is isosceles but not equilateral. Find the lengths of all three sides using the distance formula… With only two sides congruent, the triangle is isosceles, but not equilateral!

Whiteboard Problems (formative assessment) Find the area and perimeter of the figure determined by the given points: (–5, 3), (0, –1), (4, 4) Prove that the triangle determined by the points D(–3,4), O(1,0), and G(5,4) is a right triangle.

Whiteboard Problems: Quad. ABCD is a parallelogram!!! Determine the type of quadrilateral that is defined by the points A(0,0), B(7,2), C(8,12), and D(1,10). Quad. ABCD is a parallelogram!!! Find the center and radius of the circle. Center (5,1), radius = 11 Center (-1,0), radius =