P.o.D. – Factor each of the following
polynomials
1.) 25𝑥 2 − 196
2.) 8𝑥 3 + 27
3.) 21𝑥 2 − 31𝑥 − 42
4.) 6𝑥 4 + 4𝑥 3 𝑦 − 6𝑥𝑦 2 − 4𝑦 3
5.) 𝑥 2 + 5𝑥 + 6
1.) (5𝑥 − 14)(5𝑥 + 14)
2.) (2𝑥 + 3)(4𝑥 2 − 6𝑥 + 9)
3.) (3x-7)(7x+6)
4.) (3𝑥 + 2𝑦)(2𝑥 3 − 2𝑦 2 )
5.) (x+2)(x+3)
1.1 – Rectangular Coordinates
Learning Target(s): I can plot points in the
Cartesian plane; use the distance formula to
find the distance between two points; use the
midpoint formula to find the midpoint of a line
segment; use geometric formulas to model
real-life problems.
Cartesian Coordinate Plane:
- Named after Rene Descartes
- Contains 4 quadrants, two axes (x- and yaxis), origin (intersection of the axes)
Quadrant II
Quadrant I
(-,+)
(+,+)
Quadrant III
(-,-)
Quadrant IV
(+,-)
Each point (x,y) in the coordinate plane is
known as an ordered pair.
EX: Plot the points (1,-2), (4,-3), (0,0), (5,0), and
(-3,-1)
EX: Sketch a scatter plot for the following data
set:
X
Y
0
1
1
3
2
6
3
8
4
4
5
5
6
7
7
8
8
10
*Show how to sketch a scatter plot using the
graphing calculator.
The Pythagorean Theorem & The Distance
Formula:
Pythagorean Theorem: 𝑎2 + 𝑏 2 = 𝑐 2 , where a
and b are the legs of a right triangle and c is the
hypotenuse.
Distance Formula:
- This is simply an adaptation of the
Pythagorean Theorem
- 𝑑 = √(𝑥2 − 𝑥1 )2 + (𝑦2 − 𝑦1 )2
- http://www.youtube.com/watch?v=YBeXhGl6pSE
EX: Find the distance between the points (-5,7)
and (3,-4).
𝑑 = √(3 − (−5))2 + (−4 − 7)2
= √82 + (−11)2 = √64 + 121
= √185 ≈ 13.6
*Let’s write a calculator program to find
distance.
Verifying a Right Triangle:
- If a triangle is right, it will satisfy the
Pythagorean Theorem.
- Thus, we need to know the lengths of the
sides of the triangle, understanding that the
longest side must be the hypotenuse.
EX: Show that the points A(4,0), B(2,1), and
C(-1,-5) are vertices of a right triangle.
Begin by finding the distance of all 3 sides.
𝐴𝐵 = √(2 − 4)2 + (1 − 0)2 = √(−2)2 + 12
= √4 + 1 = √5 ≈ 2.236
𝐵𝐶 = √(−1 − 2)2 + (−5 − 1)2
= √(−3)2 + (−6)2 = √9 + 36 = √45
= 3√5 ≈ 6.708
𝐴𝐶 = √(−1 − 4)2 + (−5 − 0)2
= √(−5)2 + (−5)2 = √25 + 25
= √50 = 5√2 ≈ 7.071
AC is the hypotenuse. Now apply the
Pythagorean Theorem.
(√5)2 + (3√5)2 = (5√2)2
5 + 45 = 50
50 = 50
True.
The Midpoint Formula:
𝑥1 + 𝑥2 𝑦1 + 𝑦2
𝑀𝑃 = (
,
)
2
2
- Think of the midpoint as the average of two
ordered pairs.
EX: Find the midpoint of the line segment
joining the points (-6,1) and (7,4)
−6 + 7 1 + 4
1 5
𝑀𝑃 = (
,
)=( , )
2
2
2 2
*Let’s make an addition to our distance
program that will now compute midpoint.
EX: During the 3rd quarter of a recent game, the
university’s quarterback threw a pass from the
32 yard line, 40 yards from the sideline. The
pass was caught by the wide receiver on the 3
yard line, 15 yards from the sideline. How long
was the pass?
(40,32)
(15,3)
- We need to use the distance formula.
𝑑 = √(40 − 15)2 + (32 − 3)2 = √1466
≈ 38.29
EX: A local contracting company had annual
revenue of $15 million in 2003 and $17.2
million in 2005. Without knowing any
additional information, what figure would you
estimate the 2004 revenue to have been?
- We need to assume that this is a linear
relationship. Therefore, we simply need to
find the midpoint of (2003, 15) and
(2005,17.2).
2003 + 2005 15 + 17.2
𝑀𝑃 = (
,
)
2
2
32.2
= (2004,
) = (2004,16.1)
2
$16.1 million
EX: A triangle has vertices at (-1,-3), (5,1), and
(2,4). Shift the triangle two units to the left and
three units downward, and find the vertices of
the shifted triangle.
(−1, −3) → (−1 − 2, −3 − 3) = (−3, −6)
(5,1) → (5 − 2,1 − 3) = (3, −2)
(2,4) → (2 − 2,4 − 3) = (0,1)
*Study and memorize the formulas for Area,
Perimeter, Circumference, and Volume in the
box on the bottom of page 7.
EX: A beach ball has a volume of 904.32 cubic
inches. Find the radius of the sphere.
4
3
Volume of a Sphere: 𝑉 = 𝜋𝑟 3
4 3
904.32 = 𝜋𝑟
3
3
904.32 ( ) = 𝜋𝑟 3 → 678.24 = 𝜋𝑟 3
4
678.24
= 𝑟 3 → 215.89 = 𝑟 3
𝜋
3
√215.89 = 𝑟 → 5.999 = 𝑟
Upon completion of this lesson, you should be
able to:
1. Plot points on the coordinate grid
2. Utilize the Pythagorean Theorem and
the distance formula.
3. Translate coordinates of polygons.
4. Solve formulas of geometric figures.
For additional information, go to
https://www.youtube.com/watch?v=awOYuKF_LII
HW
Pg.9
3-54 3rds, 63, 65, 81-88