Parker Glynn-Adey

Math Club Geometry Training Session

Posted in Uncategorized by pgadey on 2019/01/31

These are some questions that I prepared for Math Club. The problems follow Paul Zeitz’s excellent book The Art and Craft of Problem Solving. You can find this hand-out is here:
https://pgadey.ca/teaching/2019-math-club/geometry-training-talk/geometry-training-talk.pdf (tex)

1. Advice and Suggestions

  • Draw big diagrams. Lots of them.
  • Use multiple colours to keep track on information.
  • Draw several examples with different lengths/angles.
  • Avoid coordinates.

2. Facts and Questions

Fact 1 Congurence conditions for triangles: (SAS, ASA, SSS, AAS)
Side-angle-side, angle-side-angle, side-side-side, angle-angle-side.

Question 2 Why is angle-side-side not a congruence condition?

Fact 3 If an angle {\angle A} measures a straight line, then {\angle A = \pi = 180^\circ}.

Question 4 Show that {\angle ECD = \angle ACB} using the Fact~3.

vertical-angles

Fact 5 (Transversals) {AB} and {CD} are parallel iff {\angle B = \angle D}.

transversal

Question 6 Show that the sum of the interior angles of a triangle is {\pi}. (Idea: Add a new parallel line.)

Question 7 For any triangle {\triangle ABC}, the exterior angle at {A} is the sum of the interior angles at {B} and {C}.

Question 8 (Parallelograms) Use congruence and parallels to show:

  • The opposite sides of a parallelogram have the same length.
  • The opposite angles of a parallelogram are equal, and adjacent angles are “supplementary” (sum to {\pi}).
  • The diagonals of a parallelogram bisect each other.
    (Idea: Introduce triangles.)

Question 9 (Isoceles triangles) A triangle {\triangle ABC} is isoceles if {|AB| = |AC|}. Show that: If {\triangle ABC} is isoceles then {\angle B = \angle C = 90^\circ - (\angle A)/2}.

Question 10 Let {\triangle ABC} be a triangle. Let {D}, {E}, and {F} be the midpoints of {BC}, {AC}, and {AB} respectively. Show that the lines: {DE}, {EF}, and {FD}, dissect {\triangle ABC} in to congruent triangles.

(Big Idea: Introduce parallels to {BC} and {AC} at {D}. These will create new “phantom points” on {BC} and {AC}. These new points will be very helpful because of our theory of parallelograms.)

Fact 11 (Circles and Chords) If any two are true, then all three are true.

  • The line passes through the center of the circle.
  • The line passes through the midpoint of the chord.
  • The line line is perpendicular to the chord.

Fact 12 (Circles and Tangents) A tangent to a circle is perpendicular to the radius at the point of tangency. Also, a perpendicular to a tangent line placed at the point of tangency, will pass through the center of the center of the circle.

Question 13 Suppose that {A} is outside a circle. If {AX} and {AY} are tangent to the circle, then {|AX|=|AY|}.

Question 14 (Inscribed Angle Theorem) Consider a circle centered at {O} with points {A,B,C} on the perimeter of the circle. Show that {\angle ABC = 2\angle AOC}. (Idea: Chase angles.)

inscribed-angle-theorem

Question 15 Consider a semi-circle with base {AC}.
If {B} is on the perimeter of the semi-circle, show that {\angle ABC = 90^\circ}.

Question 16 (Circumcircles) The circumcircle of a triangle {\triangle ABC} is a circle passing through {A}, {B}, and {C}.

  • Suppose that a circumcircle exists. Show that its center is the intersection point of the perpendicular bisectors of the sides of the triangle. (If a circumcircle exists, then it has a unique center.)
  • Given a triangle, any two perpendicular bisectors will intersect in a point equidistant from all three vertices. (Any triangle has a circumcenter.)
  • All three perpendicular bisectors intersect in a unique point, the “circumcenter”.

Question 17 (Incircles) The incircle of a triangle {\triangle ABC} is a circle tangent to {AB}, {BC}, and {AC}.

  • Suppose that an incircle exists. Show that its center is the intersection of any two angle bisectors.
  • Show that the intersection of any two angle bisectors is the center of an incircle and this point is unique.
  • All three angle bisectors intersect in a unique point, the “incenter”.

3. Contest Problems

Question 18 (Canada 1991) Let {C} be a circle and {P} a point in the plain. Each line through {P} that interesects {C} determines a chord of {C}. Show that the midpoints of these chords form a circle.

Question 19 (Rochester 2012) Let {\triangle ABC} be an isoceles triangle with {AC = BC} and {\angle ACB = 80^\circ}. Consider and interior point of this triangle such that {\angle MBA = 30^\circ} and {\angle MAB = 10^\circ}. Find with proof, the measure {\angle AMC}.

Question 20 (Rochester 2013) Let {\triangle ABC} be a triangle with {\angle BAC = 120^\circ}. Suppose the bisectors of {\angle BAC}, {\angle ABC}, and {\angle ACB} meet {BC}, {AC}, and {AB} at points {D}, {E}, {F}, respectively. Prove that {\triangle DEF} is a right angled triangle.

MAT B41 — Week 8

Posted in 2018 -- MAT B41, Lecture Notes, Uncategorized by pgadey on 2018/07/03
Parker comes back! Public lecture on Friday in IC 200 at 11am.

Additional resources:

Suggested Exercises:

  • Course Notes: Quadratic forms and determinants
  • 3.3 Extrema of Real-Valued Functions: 1,2,3,11,13,21,29,31,52

Past Finals:

  • Final 2015: Let f(x,y,z) = x^3 + x^2 + y^2 + z^2 - xy - zx. Find and classify the critical points of f.
  • Final 2016: Let f(x,y,z) = z^3 - x^2 + y^2 - 6yz + 4x + 4y + 6z + 1. Find and classify the critical points of f.

Notes:

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MAT B41 — Week 7

Posted in 2018 -- MAT B41, Lecture Notes, Uncategorized by pgadey on 2018/06/26
unnamed
Midterms are graded! New list of suggested exercises is available!

There is a new list of suggested exercises available here.

Suggested Exercises

  • Chapter 1 Review (p. 71): 1, 4, 5, 7, 8, 16, 18, 20
  • Chapter 2 Review (p. 145): 1, 2, 3, 5, 6, 10, 15, 25
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STLHE 2018 — Day 0 — Introduction

Posted in Teaching and Learning, Uncategorized by pgadey on 2018/06/19

I am currently at Scholarship of Teaching and Learning in Higher Education 2018 in Sherbrooke, QC. The plan is to post a short little video everyday of the conference to keep people updated on how things are going. Stay tuned!

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Mid-Course Survey

Posted in Uncategorized by pgadey on 2018/06/14

800px-Electrodynamics_vector_calculus_review_gradient.svg

The mid-course survey is available here.

MAT B41 — Week 6

Posted in 2018 -- MAT B41, Lecture Notes, Uncategorized by pgadey on 2018/06/12

Term Test is this Friday. Reading week is next week!

We have a modified class schedule this week.

  • Tuesday: Taylor Series and Review.
  • Thursday: Class Survey and Work Period, everyone is invited to bring problems to practice.
  • Friday: Term Test, will be held Friday afternoon.

mat-b41-term-test-info

Term Test Policy

From the Course Syllabus, we have the following policy:

The Term Test will be written outside of regular lecture hours. If you cannot attend reasons of creed or religion, then you will must contact Parker as early as possible to arrange for an alternative sitting. If you miss the midterm test for medical reasons, you must contact Parker within 24 hours of the test.

You will need to send a UTSC Verification of Student Illness or Injury form:

Students who miss the midterm test will be asked to provide the Verification Form and a timetable for the next five days. You will be given only one opportunity to write the make-up test.

Parker scheduled to the midterm to avoid Ramadan, but he forgot about Eid Al-Fitr. If you want to celebrate Eid with your family, instead of writing the midterm at the usual time, please contact Parker immediately.

The Week after Reading Week!

Parker will be traveling to STLHE 2018 and the IBL Workshop.

Our Tuesday evening lecture will be delivered by Ivan Khatchatourian.
ivan

Our Thursday morning lecture will be delivered by Ray Grinnell.
Ray_Grinnel

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MAT B41 — Mock Midterm!

Posted in Uncategorized by pgadey on 2018/06/08

mock-midterm

Mock Midterm was today in IC 130. Midterm Test next week!

MAT B41 — Week 5

Posted in 2018 -- MAT B41, Lecture Notes, Uncategorized by pgadey on 2018/06/05

Mock Midterm this Friday 12-2pm in IC 130. Midterm Test next week!

Several people asked about Question 7 off of Homework #3. The intent of the question was for f(u,v) to be undefined. This is different from the textbook, where the function f(u,v) is given explicitly. You may write \frac{\partial f}{\partial u} and \frac{\partial f}{\partial v} without knowing the function f(u,v).

Additional resources:

Suggested Exercises:

  • 3.1 Iterated Partial Derivatives: 1, 2, 3, 4, 5, 6, 7, 12, 14, 15, 16
  • 3.2 Taylor’s Theorem: 1, 2, 5, 9

Past Term Tests:

2016 Term Test:
Give the 4th degree Taylor polynomial about the origin of f(x,y) = e^{-xy} \arctan(x).

2015 Term Test:

  • Compute the 4th degree Taylor polynomial about the origin of f(x,y) = e^{y^2} \sin(x+y).
  • Find the linear approximation to the function f(x,y) = \frac{x+2}{4y-2} at the point (2,3) and use it to estimate f(2.1, 2.9).

Notes:

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MAT 134 — Post-Term Test #1 Survey

Posted in 2018 -- MAT 134, Math, Uncategorized by pgadey on 2018/05/31

rocky

Thank you for filling out MAT 134 Post-Term Test #1 Survey.

Here is what Parker learned about the class!

(more…)

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MAT B41 — Week 4

Posted in 2018 -- MAT B41, Lecture Notes, Uncategorized by pgadey on 2018/05/29

Homework 2 is due! Homework 3 (tex) is now available.

Midterm information:

  • Mock Term Test Friday 8 June @ 12:00-2:00 PM in IC-130
  • Real Term Test Friday 15 June @ 3:00-5:00 PM in IC-130/230

All questions regarding format, question allocation, style of test, will be addressed by the Mock Term Test. I highly recommend you attend the Mock Term Test, because it will be the perfect preparation for the Real Term Test. More information will be available early next week.

In order to prepare for the Term Test, I recommend attempting the suggested exercises.
Additional resources:

Suggested Exercises:

  • 2.3 Differentiation: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 19, 22, 24, 25 (Do as much as possible.)
  • 2.5 Properties of Derivatives: 2, 3, 6, 14, 17
  • 2.6 Gradients and Directional Derivatives: 1, 4, 6, 10, 11, 19

Past Term Tests:

2016 Term Test: Let f: \mathbb{R}^2 \rightarrow \mathbb{R} be given by f(x,y) = (x+y)/x^2.

  1. Find an equation for the tangent plane to the graph z = f(x,y) at the point (2,3,f(2,3)).
  2. Find the direction of maximum increase in f at the point (2,3). What is the rate of maximum increase?

2015 Term Test:

  1. Determine the rate of change in f(x,y) = y - x^2 + z^2 as you move from (-1,0,2) towards (2,4,2).
  2. Compute the directional derivative of f(x,y,z) = x^2y^3z^2 at the point (2,1,-1) in the direction of the upward normal for the plane 2x+y-2z = -7.

Notes:

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