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

2. 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

3. 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

4. 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)

5. 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)

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

7. 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)

8. 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

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

10. 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:

11. 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!

12. 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.

13. 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 =