Coordinate Geometry Independent Study
Outline and Assessment
Outcomes:
E5 apply inductive reasoning to make conjectures in geometric situations
D1 develop and apply formulas for distance and midpoint
E7 investigate and make and prove conjectures associated with chord properties of circles
E11 write proofs using various axiomatic systems and assess the validity of deductive arguments
E4 apply properties of circles
E15adv solve problems involving the equations and characteristics of circles and ellipses
Tasks / Marks1. Understand the language
Check your understanding of words in the word list using the sites or a math dictionary
2. Investigate Slope, Distance and Midpoint
Be sure you understand how each formula is developed and used. Correct form is
important in the presentation of solutions.
3. Practice using formulas
Do: Page 222 #1, 3
Page 226 #10, 11, 12, 13, 14 (hand in on graph paper)
Page 230 #28, 29
Check your answers (text page 327)
Based on completeness. / 5
4. Complete the booklet.
Based on completeness.
Four questions (teacher’s choice), marked for mathematical content and presentation. / 5
18
5. Cover page. / 2
Total / 30
Timeline:
Assigned: April 27, 2012
Due: May 14, 2012
Quiz: May 14, 2012 /
The final mark for this assignment will be: Assignment 50% and Quiz 50%. In the event that a student scores 80% or better on the quiz, the assignment will automatically be valued at 30/30. You must complete the Independent study to qualify for full value!
Coordinate Geometry Independent Study
1. Word list:
Be sure you fully understand the meanings of the following words. / If you need clarification, check these sites or a math dictionary:Altitude
Angle
Chord
Congruent
Converse
Diameter
Equidistant
Equilateral
Isosceles triangle / Line
Line segment
Polygon
Quadrilateral
Radius
Ray
Rhombus
Scalene triangle
Trapezoid / http://www.teachers.ash.org.au/jeather/maths/dictionary.html dynamic, fun, a little childish
http://users.adelphia.net/~mathhomeworkhelp/ more sophisticated
http://thesaurus.maths.org/ very thorough – provides background
http://www.intermath-uga.gatech.edu/dictnary/homepg.asp thorough - check out the menu on the left for more details
http://www.infomath.com/html/geometryglossary1.asp glossary of geometry terms
2. Developing Formulas: Use the websites and/or the text to help you understand and use the formulas. Then do the indicated questions from the text and check your answers (see text)
Websites: / Text:Slope:
http://www.purplemath.com/modules/slope.htm examine the entire page
http://mathforum.org/cgraph/cslope/ very thorough
http://www.math.com/school/subject2/lessons/S2U4L2GL.html
http://regentsprep.org/Regents/math/math-topic.cfm?TopicCode=glines visit the pages marked L and P
http://cs.selu.edu/~rbyrd/math/slope/ simple, basic, self test
http://www.explorelearning.com/index.cfm?method=cResource.dspDetail&ResourceID=87 interactive, steps for calculating slope. Click on Launch Gizmo. Click on Exploration Guide for directions if you need them.
http://math.usask.ca/~maclean/101/Review/Printables/BW/Lines.pdf slopes and equations of lines examples
my GSP activity / Investigation 3
Page 222 – 223
Do: Page 222 #1, 3
Distance formula:
http://regentsprep.org/Regents/math/distance/Ldistance.htm
http://regentsprep.org/Regents/math/math-topic.cfm?TopicCode=distance go to L and P pages
http://www.purplemath.com/modules/distform.htm uses Pythagorean theorem; gives examples
http://www.learner.org/channel/courses/learningmath/geometry/session6/part_c/distance.html from Pythagoras to the distance formula
http://cs.selu.edu/~rbyrd/math/distance/ simple, basic, self test
http://www.explorelearning.com/index.cfm?method=cResource.dspDetail&ResourceID=183
click on Launch Gizmo / Focus C
Page 224 - 226
Do: Page 226 #10, 11, 12, 13, 14
Midpoint:
http://www.mathsnet.net/dynamic/jsp6.html what is meant by midpoint?
http://www.purplemath.com/modules/midpoint.htm look at the examples showing how it is used
http://cs.selu.edu/~rbyrd/math/midpoint/ simple, basic, self test
Do: Page 230 #28, 29
Summary of formulas
Slope / Midpoint / Distance= / = / =
=
Parallel
(use || for “is parallel to”)
Perpendicular
(use for “is perpendicular to”) / If two lines are parallel, their slopes are equal.
If two lines are perpendicular their slopes are negative reciprocals. The product of the slopes is -1.
Equation of a line:
Slope y-intercept form / Slope Point FormIf m = slope and b = y-intercept then the equation is
y = mx + b / If (x1, y1) is a point on the line and m = slope then the equation is
y – y1 = m(x - x1)
A) What are the essential features of a rectangle that are captured using the
coordinates shown in the diagram.
B) Why is the diagram general enough to apply to ALL rectangles?
C) Prove, using the indicated coordinates, that the diagonals of a rectangle bisect
each other.
D) Are the diagonals of a rectangle perpendicular? / General proofs:
In order to apply the results of a proof to all figures that satisfy a particular condition (rectangle), you must select coordinates that capture the essential conditions but are still general enough to apply to all such figures.
Rectangle PQRS has coordinates P(0, 0), Q(0, b), R(a, b), and S(a, 0),
12. A circle contains two chords, GH and JK, with
endpoints G(-10,4), H(-2,16), J(8,16), and K(16,4).
A) Find the midpoint of each chord. Mark these midpoints
on your diagram.
B) Find the slope of each chord.
C) Find the slope of the perpendicular bisector to chord
GH, and the slope of the perpendicular bisector to JK.
Draw the new lines.
D) Determine the equation of each new line in part (c).
E) Find the point of intersection, C, of these two lines.
F) Verify that C is the centre of the circle by determining
distances CG, CH, CJ, and CK.
G) What is the radius of the circle?
H) Determine 3 more points on the circle. /
T(0, -3).
A) Show thatΔCAT is a right angled triangle.
B) Show that the midpoint of the hypotenuse, M, is the centre of a circle
that passes through the vertices of the triangle. /
15. General proof: Prove that the diagonals of a parallelogram bisect each
other. /
16. Prove that the diagonals of a square are congruent and that they are
perpendicular bisectors of each other. /
17. A triangle has vertices A(0,0), B(4,6), and C(8,2).
A) Find the coordinates of the midpoint of each side?
B) Determine the equation of each line joining a midpoint to the opposite vertex?
C) Verify that the lines in part (b) intersect at a single point. /
D) Determine the equations of the altitudes.
E) Verify that the altitudes intersect at a single point.
18. A circle with centre C(-1, 2) has chords BL and UE with endpoints B(-6, 14),
L(-13, -3), U(-6, -10), and E(11, -3).
A) Prove that the chords are congruent.
B) Prove that the chords are equidistant from the centre of the circle. /