Transcript Coordinate Geometry

Coordinate Geometry
Time is running out!!!!!
 The Tuesday after Thanksgiving break (11/30) is the
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last day to turn in any work from Unit 3 – Geometry
The Thursday after Thanksgiving break (12/2) is the
last day to turn in any work from Unit 4 – Chance of
Winning (Statistics and Probability)
The following Tuesday (12/7) is the last day to turn in
any work from Unit 5 – Algebra in Context.
The following Thursday (12/9) is the last day to turn in
any work from Unit 6 – Coordinate Geometry
The following week is finals.
 I have put links to usatestprep.com and the
Department of Education for your use to review for the
EOCT and GHSGT
Standards
MM1G1:
Students will investigate properties of
geometric figures in the coordinate plane.
Determine the distance between two points.
Determine the distance between a point and a line.
Determine the midpoint of a segment.
Understand the distance formula as an application
of the Pythagorean Theorem.
Use the coordinate plane to investigate properties
of and verify conjectures related to triangles and
quadrilaterals.
When will we ever use rational
and radical functions?
THIS WEEK!!!
Disclaimer:
 Please be patient and nice to yourself. It is my
intention to use this section as a review of topics we
have covered. This means, you will be required to
remember (or learn) what was covered in the past. I
expect you to bring your books, look up the
information you may not remember, and work in small
groups to figure out the problems. We are reviewing
for the EOCT and final.
 If you do not want to do this work, please bring
something else to study quietly. I will not be in the
mood for you to disrupt the class.
Pythagorean Theorem – 4.1
 If you have a right triangle, then the square of the
hypotenuse equals the sum of the square of the legs.
a
c
b
a2 + b2 = c2
 Proof
Pythagorean Theorem – 4.1
 Determine the length of the hypotenuse of a right
triangle if the legs are 7” and 13” long\
 Solution:
13   7  
2
2
218
 Determine the length of the second leg if the
hypotenuse is 15” and one leg is 9”.
 Solution:
15  9  
2
2
144  12
Distance Formula – 1 Dimension
 The distance between two numbers in one dimension
(on a number line) is the difference of the values of the
points.
 Example: Find the distance between 9 and 4.
 9–4=5
 Example: Find the distance between 13 and -3
 13 – (-3) = 13 + 3 = 16
Distance Formula – 2 Dimensions
 The distance formula in two dimensions is straight
from Pythagorean Theorem
 The Cartesian Coordinate System is two number lines
placed at right angles to each other, so:
 Make a triangle using the points as the end of the
hypotenuse and the legs parallel to the x-axis and the
y-axis
 Pythagorean Theorem says that the square of the
distance between the two points (the hypotenuse)
equals the sum of the squares of the lengths of the legs
 The length of each leg is calculated just like in onedimension
Distance Formula – 2 Dimensions
d
x2  x1 2   y2  y1 2
 Example: Find the distance between (-1, 5) and (2, 9)
 The square of the distance is:
2  (1)  9  5  3  4
2
 So the distance is:
2
25  5
2
2
 9 16  25
 What is the largest beam you can cut out of a 12 inch
diameter log if the height has to be three times the
width?
x
3x
Distance Formula
 Find the diagonal of a cube with side length of x
Midpoint in One Dimension
 What is the midpoint between 5 and 7? How did you
determine it?
 The midpoint is found by taking the average of the
point.
 Midpoint = x1  x2
2
5  7 12
 Midpoint =

6
2
2
Midpoint in Two Dimensions
 Find the midpoint in two dimensions by calculating
the midpoint in each (x and y) dimension.
 Midpoint =  x1  x2 y1  y2 
,


2 
 2
 Example: Find the midpoint between (5, 3) and (-2, 7)
 Midpoint =  5  (2) 3  7   3 10 
,

   ,   1.5,5
2  2 2 
 2
Homework
 Pg 195, # 3 – 21 by 3’s and 22 – 25 all (11 problems)
Warm-Up
 Given the mid point of a line is (2, 3) and one end is at
(-2, 5), find the other end point.
 Find the length of the line segment with the endpoints
at (2 1/2, 3 2/3)and (1 1/3, 1/5).
Slope
 Rise over run, change in y over change in x
y2  y1
slope 
x2  x1
 Parallel lines have the same slope
 The products of the slopes of perpendicular lines
equal -1  they are upside down and negatives of
each other.
Slope
 Example: Find the slope between (10, -1) and (2, 3)
3  (1)
4
1
slope 


2  10
8
2
 Example: Find the slope of the line perpendicular to
the line through the above two points.
Perpendicular slope =  2
1
System of Equations
 We can use the idea of f(x) = g(x) to find the
intersection of two lines.
 Set f(x) = g(x) and solve for the value of x that satisfies
the equation.
 Substitute the value of x into either equation to find
the value of y for that value of x.
 Substitute the (x, y) you found into the second
equation for a check.
Systems of Linear Equations
 Example: Determine where the following two lines
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intersect: f(x) = 3x – 8 and g(x) = -4x + 6
Let f(x) = g(x), or 3x – 8 = -4x + 6
x=2
f(2) = -2, so the intersection is (2, -2).
Substituting into g(x) gives g(2) = -4(2) + 6 = -2, check.
Systems of Linear Equations
 Find the intersection of the following lines:
 y = 3x + 1 and y = -x + 5
 Solution: (1, 4)
 3x + y = 4 and x – 2y = 6
 Solution: (2, -2)
Distance from Point to Line
 The distance between a point and a line is the
perpendicular distance from the line to the point.
 Steps of finding the perpendicular distance:
 Find the slope of the original line
 Determine the slope of the perpendicular line.
 Determine the equation of the perpendicular line going
through the point.
 Determine where the two lines intersect by solving the
system of equations.
 Find the distance between the two points via
Pythagorean Theorem.
Distance from Point to Line
 Example: Find the distance from point (9, 1) to the line
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f(x) = 5x + 8
What is the slope of f(x)? 5
What is the slope of the line perpendicular to f(x)? -0.2
The point (9, 1) must be on the perpendicular line.
What is the equation of the line g(x) with a slope -1/5
going through (9, 1)? y = -0.2x + 2.8
What is the intersection of f(x) and g(x)? (-1, 3)
What is the distance between (9, 1) and (-1, 3)? 2 26
Homework
Page 232, # 19 – 24 all (6 problems)
Coordinate Proofs
 We will be proving geometric theorems by placing
geometric shapes on the Cartesian Coordinate plane
and using the above formulas.
 Judicial placement of the shapes can simplify our
calculations.
 We often (not always) place one vertex at the origin,
and one side on the x-axis.
Communicate
In coordinate geometry proofs, why is it possible to
place one vertex of the figure at the origin and one
side along the x-axis?
2. In a coordinate geometry proof about triangles,
could you place one vertex at the origin, one on the xaxis, and one on the y-axis? Why or why not?
3. In a coordinate geometry proof, what do you need to
show to prove that two lines are parallel? That two
lines are perpendicular?
4. In a coordinate geometry proof, what do you need to
show to prove that two segments bisect each other?
1.
 We do not always put the vertex at the origin and one
side on the x-axis
Prove via Coordinate Geometry
 The opposite sides of a parallelogram are congruent
 The diagonals of a square are perpendicular to each
other
 The diagonals of a rectangle are congruent
 If the diagonals of a parallelogram are perpendicular,
then the parallelogram is a rhombus.
 CHALLENGE: The medians of a triangle intersect at a
single point. (HINT: Find the equations of the lines
containing the medians)
Prove via Coordinate Geometry
 The diagonals of a square are perpendicular.
 The three segments joining the midpoints of the sides of an
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isosceles triangle form another isosceles triangle
The diagonals of a parallelogram bisect each other
The segments joining the midpoints of the sides of an
isosceles trapezoid form a rhombus.
If a line segment joins the midpoints of two sides of a
triangle, then it is parallel to the third side.
If a line segment joins the midpoints of two sides of a
triangle, then its length is equal to one-half the length of
the third side.
The line segments joining the midpoints of the sides of a
rectangle form a rhombus