Coordinate Geometry

Time is running out!!!!! The Tuesday after Thanksgiving break (11/30) is the 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. ac b a 2 + b 2 = c 2 Proof

Pythagorean Theorem – 4.1 Determine the length of the hypotenuse of a right triangle if the legs are 7” and 13” long\ Solution: Determine the length of the second leg if the hypotenuse is 15” and one leg is 9”. Solution:

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) = = 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 one- dimension

Distance Formula – 2 Dimensions Example: Find the distance between (-1, 5) and (2, 9) The square of the distance is: So the distance is:

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 =

Midpoint in Two Dimensions Find the midpoint in two dimensions by calculating the midpoint in each (x and y) dimension. Midpoint = Example: Find the midpoint between (5, 3) and (-2, 7) Midpoint =

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 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) Example: Find the slope of the line perpendicular to the line through the above two points. Perpendicular slope =

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 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 f(x) = 5x + 8 What is the slope of f(x)? What is the slope of the line perpendicular to f(x)? 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)? What is the intersection of f(x) and g(x)? What is the distance between (9, 1) and (-1, 3)? y = -0.2x (-1, 3)

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 1. 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 x- axis, 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?

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 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