## Math Club Geometry Training Session

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 1Congurence conditions for triangles: (SAS, ASA, SSS, AAS)

Side-angle-side, angle-side-angle, side-side-side, angle-angle-side.

Question 2Why is angle-side-sidenota congruence condition?

Question 4Show that using the Fact~3.

Fact 5 (Transversals)and are parallel iff .

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

Question 7For any triangle , the exterior angle at is the sum of the interior angles at and .

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 ).
- The diagonals of a parallelogram bisect each other.

(Idea: Introduce triangles.)

Question 9 (Isoceles triangles)A triangle is isoceles if . Show that: If is isoceles then .

Question 10Let be a triangle. Let , , and be the midpoints of , , and respectively. Show that the lines: , , and , dissect in to congruent triangles.

(Big Idea: Introduce parallels to and at . These will create new “phantom points” on and . 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 13Suppose that is outside a circle. If and are tangent to the circle, then .

Question 14 (Inscribed Angle Theorem)Consider a circle centered at with points on the perimeter of the circle. Show that . (Idea: Chase angles.)

Question 15Consider a semi-circle with base .

If is on the perimeter of the semi-circle, show that .

Question 16 (Circumcircles)The circumcircle of a triangle is a circle passing through , , and .

- 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 is a circle tangent to , , and .

- 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 be a circle and a point in the plain. Each line through that interesects determines a chord of . Show that the midpoints of these chords form a circle.

Question 19 (Rochester 2012)Let be an isoceles triangle with and . Consider and interior point of this triangle such that and . Find with proof, the measure .

Question 20 (Rochester 2013)Let be a triangle with . Suppose the bisectors of , , and meet , , and at points , , , respectively. Prove that is a right angled triangle.

## MAT B41 — Week 8

**Parker comes back! Public lecture on Friday in IC 200 at 11am.**

**Additional resources:**

- Khan Academy on multivariate maxima and minima.
- Kristal King on local extrema.

**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 . Find and classify the critical points of .*Final 2016*: Let . Find and classify the critical points of .

**Notes:**

## MAT B41 — Week 7

**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

## STLHE 2018 — Day 0 — Introduction

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!

## MAT B41 — Week 6

**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.

**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.

Our Thursday morning lecture will be delivered by Ray Grinnell.

## MAT B41 — Week 5

**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 to be undefined. This is different from the textbook, where the function is given explicitly. You may write and without knowing the function .

**Additional resources:**

- Wikipedia on Clairaut’s theorem
- Khan Academy on Second Partial Derivatives
- Krista King on Second Partial Derivatives
- PatrickJMT on Second Partial Derivatives
- ThreeBlueOneBrown on Single Variable Taylor Series
- Mark Ancliff on Multivariate Taylor Series

**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 .

*2015 Term Test*:

- Compute the 4th degree Taylor polynomial about the origin of .
- Find the linear approximation to the function at the point and use it to estimate .

**Notes:**

## MAT 134 — Post-Term Test #1 Survey

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

Here is what Parker learned about the class!

## MAT B41 — Week 4

**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:**

- Khan Academy on partial derivatives.
- PatrickJMT on partial derivatives.
- KristaKing on partial derivatives.
- Eugene Khutoryansky explains gradients and partial derivatives. (Beautiful animation!)
- Educational Videos explains gradients and partial derivatives.

**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 be given by .

- Find an equation for the tangent plane to the graph at the point .
- Find the direction of maximum increase in at the point . What is the rate of maximum increase?

*2015 Term Test*:

- Determine the rate of change in as you move from towards .
- Compute the directional derivative of at the point in the direction of the upward normal for the plane .

**Notes:**

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