Class on 2/27/09

Read the example problem shown on page 244 and then do problems 4-8 on page 245 of your textbook.

(Note: This work will obviously be done in your notebooks.)

Class on 2/25/09

[Please make sure to read the entire post!]

Read pages 239 to 240 of the textbook. We proved the theorem and its three corollaries that follow in class today. So, once you have proved the theorem and read about mean proportionals you should try to prove the corollaries by using the proportions of the similar triangles you prove in the theorem. Make sure to prove each case and understand why it works and then read the words of the corollaries to make sense of the what you have proved. The words of the corollaries actually help remember the relationships.

Next, do problems 5, 7, 9, 15 and 16 on page 241 using the theorem and its corollaries that you just read (the questions are pretty easy and should not take you time). Question 16 is both interesting and crucial at the same time. If you cannot prove the statement in that question, then don't worry about it because if you turn to pages 242-243, then this famous theorem and its converse have been proven for you and you can read the proof and understand how it is done. (If you are Connor Miksch and have proved this theorem another way, you are still required to prove it this way and understand it the books way!)

You must complete the work above today so that you can do the work for tomorrow which includes a reading packet that the sophomores already got today to go home and read. You will have to pick up your reading packet (titled The Pythagorean Theorem) from the front desk before, during or after school tomorrow. Make sure to pick it up and read it as you will be assessed on this reading on Monday (reading quiz). The reading will provide you with important background information and context regarding the theorem and will also paint some of the landscape that will be necessary for trigonometry later. Do not save it for the weekend as you will have homework over the weekend.

Note: You are not required to take notes on this reading.

Class on 2/24/09

Do problems 1, 4, 5, 12 & 15 on page 235 of your textbook. You may use this post to discuss any of the questions.

Solution to #12 on Page 234

For this problem you must remember the theorem that states that the altitudes of similar triangles are proportional. To keep he post short I have summarized the solution. The diagram to te right is what I will be referring to for this solution. Some points were in the description in the question and others are introduced as a result of the auxiliary lines.

You know that QDC is similar to QAC since their angles are equal. Thus, AQ/DQ = BQ/CQ = AB/DC. The ratio of AB/DC is 3/2 from the given. Moreover, the altitudes of similar triangles are also proportional and so JQ/KQ = AQ/DQ. Therefore, JQ/KQ = 3/2. We are first interested in finding the length of JQ, which consists of JK+KQ. JK is 1.5 inches since the distance between DC and AB is 1.5 inches. Thus, 1.5 +KQ = JQ. Then, (1.5 + KQ)/KQ = 3/2. You can now solve this equation for KQ. When you have KQ then you can know JQ, which is the distance of Q from AB.

Now consider my auxiliary line through P that is perpendicular to both AB and DC and serves as the distance between AB and DC just as JK does. Now due to alternate interior angles formed by the dagonals and equal opposite angles you know that ABP and CDP are similar triangles. Also, you will find that PY and PX are altitudes for those triangles, respectively. If the triangles are similar then the altitudes are similar. Thus, AB/DC = PY/PX. AB/DC = 3/2 and so PY/PX = 3/2. We know that PY+PX = 1.5 inches from the given. So, you have two equations and two variables. Now all you have to do is find PY.

Homework due 2/17/09

Start by doing exercises 4-10 on page 224 and then from Exercises [A] on pages 225-226 do problems 1-4, 7 and 9. Next, read pages 226-228 and make sure to understand the proofs for the theorems regarding similar triangles discussed on those pages. You will be required to take notes on this reading. In your notes please try and re-write, in paragraph form, the proofs for the theorems.

[Note: For those students that are absent on Friday, Feb 13th, you can download the handout we will have done in class from the class conference folder and you will be expected to complete the handout and then do the homework that is assigned in this post.]

Quiz on Friday, February 20th

You will have a quiz on Ratio and Proportions, and whatever we will have done with Similar Figures till then.

Homework due 2/13/09

Read Case B of the theorem about angle bisectors on page 217 and understand the proof for it, making sure to read why Case B does not work for when the sides of the triangle are equal. Then do problems 3, 5 and 10 on pages 218-219.

Homework due 2/11/09

Read pages 212-214 and then read the blog post from last semester that talked about the issue of "number vs. magnitude". You can search it in the top left hand corner of the blog where there is a search field.

Homework due 2/10/09

Do exercises 3, 5, 6 & 9-12 on pages 208-209.

Tips for Complex Proofs

So, we may have reached a stage where just crafting a proof in our mind and then putting it down on paper is not the best way for more involved and complicated proofs. In essence, we have come to what geometry is all about; exploration! To make the process of discovery more effective, try some of the suggestions below.

1. Draw a reasonably large diagram so that relationships between lines and angles are easy to spot and label and that auxiliary lines find adequate space to be drawn.

2. Label all that you can on the diagram and if you're afraid of clutter then consider using some alternatives for marking congruent angles and lines. (Jack's suggestion of using roman numerals is effective!)

3. Start forming a list of relationships you do see as being true. If possible, jot down some reasons so that the task of writing a formal proof is not too daunting. The list is important because often the diagram cannot house all the relationships and the list could help you establish relationships, especially algebraic ones.