P.2: Absolute Value, P.2: Absolute Value, Distance Formula, Distance Formula, Midpoint Formula, Midpoint Formula,

Equations of CirclesEquations of Circles

First, recall the Cartesian Plane:

(also called the rectangular coordinate system)

x

y Ordered pair:

P(3, 1)

x-axisx-axis

y-axisy-axis

P(3, 1)

x-coordinatex-coordinate

y-coordinatey-coordinate

originorigin

FirstQuadrant

SecondQuadrant

ThirdQuadrant

FourthQuadrant

Absolute Value

π

…tells magnitudeof a number…

The absolute value of a real number a is:

|a|=

a, if a > 0

Not

atio

n!

Not

atio

n!

–a, if a < 0

0, if a = 0

Ex: |5| = 5

Ex: |–3| = 3

Ex: | – 6| = 6 – π

Distance FormulaLet 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:

d = (x – x ) + (y – y )2 22 21 1

(derived via the (derived via the Pythagorean TheoremPythagorean Theorem))

1 1 2 2

Midpoint Formula

The midpoint of the line segment with endpointsa and b is:

The midpoint of the line segment with endpoints(a, b) and (c, d) is:

a + b2

a + c b + d

2 2,

(Number Line)(Number Line)

(Coordinate Plane)(Coordinate Plane)

Guided PracticeFind the distance between the points:

–2, –11 (–3, 1), (7, –3)

d= 9 d = 2 29

Guided Practice

Find the midpoint of the line segment with thegiven endpoints:

2.3, –10.4

Midpt = – 4.05 Midpt = (–½, 5)

(5, 8), (–6, 2)

What is a circle???

The set of points in a plane at a fixeddistance (radius) from a fixed point (center)

x

y

r

(h, k)

(x, y)Let’s use the distance formulato find the radius:

2 2( ) ( )r x h y k

Then, what happens if we square both sides…

Standard form equation for a circleStandard form equation for a circle

2 2 2( ) ( )x h y k r

Guided Practice:

Find the standard form equation of the circle.

Center (–3, 6), radius 9

Center (0, 2), radius

2 2( 3) ( 6) 81x y 3 2

2 2( 2) 18x y

Guided Practice:

Describe the set of real numbers that satisfy:

2 3x

1 5x

This expression means that the distance from xto 2 must be less than 3…

Thus, x must be between –1 and 5:

Guided Practice:

Prove that the triangle determined by the points (3,0), (–1,2), and (5,4) is isosceles but not equilateral.

With only two sides congruent, theWith only two sides congruent, thetriangle is isosceles, but not equilateral!triangle is isosceles, but not equilateral!

Find the lengths of all three sides using the distance formula…

2 5

2 5

2 10

Whiteboard Problems (formative assessment)

Find the area and perimeter of the figuredetermined by the given points:

(–5, 3), (0, –1), (4, 4) 20.5

2 41 82 21.86

Area

Perimeter

Prove that the triangle determined by the pointsD(–3,4), O(1,0), and G(5,4) is a right triangle.

32

32

64

DO

OG

DG

2 2 2DO OG DG

Determine the type of quadrilateral that is defined bythe points B(0,0), A(7,2), I(8,12), and N(1,10).

Quad. BAIN is a parallelogram!!!Quad. BAIN is a parallelogram!!!

Whiteboard Problems:

Find the center and radius of the circle.2 2( 5) ( 1) 121x y Center (5,1), radius = 11

2 2( 1) 80x y Center (-1,0), radius = 4 5

Homework:

p. 20-22 11-17 odd, 23-27 odd, 41-51 oddQuiz P1-P2 tomorrow. If you have questions over the homework you must see me before the quiz (which usually means before school 7:15-8:00). We won’t have time during class to review the HW.