Coordinate Geometry

Coordinate Geometry is a cross between algebra and geometry.  So, I’ve included it in both the algebra and geometry parts of my course.  That way, students who are skipping around in the course will be sure to get it before they get to algebra functions which requires an understanding of coordinate geometry.

Words of “Wiz-dom”-This topic sounds worse than it is. First, you need to compute distances on horizontal and vertical number lines and their diagonals. No problem. It’s as simple as subtracting numbers and using the Pythagorean Theorem (a2+b2=c2). Second, you have to calculate “slope.” If you can subtract and divide, you can do this too. Let’s get started with a quick review. Always be on the lookout for “hidden” right triangles.

Whenever you determine the position of a coordinate point, be sure you remember that the x position is written before the y position (x,y). Also keep in mind each quadrant has its own signs. Starting with Quadrant I at the top right, the signs are (+,+) and going counter-clockwise to Quadrant II (-,+), to Quadrant III (-,-), and finishing in Quadrant IV (+,-).

Calculating Distances:

  • For horizontal lines, subtract the x coordinates.

  • For vertical lines, subtract the y coordinates.

  • For “diagonal” lines, treat it as the hypotenuse of a right triangle.

Calculating Slopes:

“Slope” tells you how fast a line went up or down as it is drawn from left to right. There are two special cases: Zero (0) Slope means it’s a horizontal line (It doesn’t go up or down!) and “undefined” slope means it goes straight up and down. All other slopes are calculated by dividing how far the line goes up or down (change in y) divided by how far the line goes from left to right (change in x). This is sometimes called the rise (or fall) divided by the run. If the line is going up, it’s a positive slope. If it’s going down, it’s a negative slope.

What are the coordinate positions of:
A
B
C
D

What are the lengths of:
AC
CF

What are the slopes of:
AC
CD
CF

Words of “Wiz-dom” on Basic Coordinate Geometry

What is the special shape of the lines formed by the coordinates at C, D, and F? Answer  #1

What is the perimeter of CDF? Answer  #2

What is the area of CDF? Answer  #3

What is the special shape of the lines formed by the coordinates at C, D, and A? Answer  #4

What is the slope of AD? Answer  #5

What is the perimeter of ACD? Answer  #6

What is the area of ACD? Answer  #7

Sample Questions

14. For line PQR, P has coordinates (2,4) and R has coordinates (2,12). If the ratio of the lengths of PQ to QR is 3:1, what is y-coordinate of Q?

Answer to Question #14

7.  The Wizard’s owl, Orville, flew 5 miles due east from the Castle of Wiz-dom and then turned due north and went 12 more miles. What is the shortest distance in miles back to the castle?

(A) 8.5
(B) 12
(C) 13
(D) 15
(E) 17

Answer to Question #7

15. Phalange, my falconer, took Beak ‘N Talons, my favorite hunting falcon, out for some exercise. When released, he flew 300 meters west and then turned due north and flew another 400 meters. At that point, he abruptly soared 1200 meters straight up! When he reached that high point, what is the shortest flight in meters for Beak ‘N Talons back to the falconer?

(A) 500
(B) 1,200
(C) 1,250
(D) 1,300
(E) 1,500

Answer to Question #15

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