MathNotations

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I probably should start a new word game called find all the typos! 'Simpel' and rightt' are pretty impressive variations. I guess if you can't spell write rite, then you're not alright! Well, I'm trying to correct these but once they go to a reader, the mistakes are out there for all to see! Haste makes --------------.

Just another one of those square and right triangle problems. Yes, I do have a passion for these types of geometry questions (I've published similar ones previously) and you probably will recognize this one from your own experience. The difference of course is that an investigation takes students from particular cases to a general formula which is related to the harmonic mean of the legs of the right triangle.



Part I
Find the length of the side of the square in Fig. I.
Comments:
This one is eminently guessable and doesn't require the use of similar triangles. Encourage students to justify their conjecture that D and F are midpoints.

Part II
Do the same for Fig. II.
Comments: Ok, the answer is 12/7. You may find students trying a Pythagorean approach, guessing midpoints or assuming special angles. Most students do not look for similar triangle solutions unless they have considerable experience with this or the problem is assigned in that unit!

Part III
Of course, we will now ask students to generalize the result:
If the legs of the right triangle have lengths a and b, show that the side of the indicated square (inscribed square, largest inscribed square with sides parallel to the legs, however you want to describe it!) has length ab/(a+b).

Posted by Dave Marain at 8:18 AM 12 comments

Labels: geometry, investigations, right triangles, similar triangles, squares

The connections between geometry and other rich areas of mathematics are boundless. Here is a fairly straightforward set of problems that can be explored as far as your eye and mind can see. On the surface, we have three rectangles each of which has a half-diagonal of length 6. Students can be asked to find the area of each without using any trigonometry. On a deeper level, one can ask students to explain or prove why the square has the greatest area for a given
diagonal length. This is straightforward using the well-known trig formula for area of a triangle, K = (1/2)absin C, however the challenge here is to use non-trig methods (although the student can use special right triangles) to compute the areas and demonstrate the maximum. The maximum piece is more sophisticated and the idea of bringing this in before precalculus and calculus has many benefits.

The instructor might begin by asking students to draw any rectangle with a diagonal of 12. How many such rectangles could there be? Which one would appear to have the greatest area? Ok, now let's explore a few special cases.

This problem allows the creative student to devise a visual way of explaining the maximum. It also allows the instructor to bring in the Arithmetic Mean-Geometric Mean Inequality for enrichment. So many methods and approaches are possible...

Posted by Dave Marain at 6:39 AM 2 comments

Labels: AM-GM Inequality, geometry, maxima/minima problem, rectangles, squares

Posted by Dave Marain at 3:56 PM 4 comments

Labels: deductive reasoning, geometry, quadrilaterals, squares

Since I only received one comment from the squares problem , I guess readers were either bored by this question or are awaiting my solution(s)!

Solution I: Divide side AB into segments of lengths 2+x, 2-x. Here, x can be any real between -2 and 2. Similarly, divide BC into segments of lengths 2-y and 2+y. One can demonstrate that the order here is irrelevant. Then the product of the areas of either pair of non-adjacent rectangles can be expressed (after rearrangement of factors) as
(2+x)(2-x)(2+y)(2-y) = (4-x²)(4-y²).
From the restrictions on x and y, it follows that x² is greater than or equal to zero and less than 4. Similarly for y.
Therefore, (4-x²)(4-y²) ≤ 4⋅4 or 16.
This also demonstrates that the maximum product occurs when x=0 and y=0! QED

Solution II (using the Arithmetic-Geometric Mean Inequality): We will prove a general result for squares of side m. This will be forthcoming and there may be a visual surprise! Stay tuned!

Posted by Dave Marain at 8:07 AM 0 comments

Labels: geometry, investigations, optimization, squares

Before announcing the thousands (or less!) winners of the Name That Mathematician Challenge, I came across a problem about dissecting a square ABCD with lines PQ and RS which are parallel to the sides of the square. (see diagram).

Naturally, I decided to make it into a deeper investigation. Students and/or readers will be asked to find the maximum value of the product of the areas of either pair of non-adjacent rectangles formed. There are many approaches here, one of which uses the famous Arithmetic Mean-Geometric Inequality. As usual you will work from the particular to the general, beginning with a specific value for the sides of ABCD.

STUDENT/READER INVESTIGATION - PART I
The given conditions about the diagram are given above.

For Part I, we will assume each side of the square has length 4.

(1) (Particular) If AP = 3 and RC = 2, determine the product of the areas of APTS and RTQC. Do the same for the other pair of non-adjacent rectangles formed. Do you believe this product is the maximum possible as we vary the positions of segments PQ and RS?

(2)(General) Show that the product of the areas of either pair of non-adjacent rectangles formed is less than or equal to 16. For example the product of the areas of APTS and RTQC is ≤ 16.

Notes:
(1) Do you think many students would guess what the configuration would be for the maximum product to occur? Is proving the conjecture much more difficult?
(2) The challenge here is to find an effective use of variables to denote the segments. There are many possibilities, some much more efficient than others.
(3) I will add additional parts to this challenge after receiving comments on Part I. How would you generalize this result further? More interestingly, there is a way to prove Part I using the AM-GM Inequality?

Posted by Dave Marain at 5:58 AM 2 comments

Labels: AM-GM Inequality, geometry, investigations, optimization, squares

Two of the opposite vertices of square PQRS have coordinates P(-1,-1) and R(4,2). (a) SAT-Type: Find the area of PQRS.
SAT Level of difficulty: 4-5 (i.e., moderately difficult to difficult).
Note: For standardized tests, in particular, students are encouraged to learn the special formula for the area of a square in terms of its diagonal.


Now for a more significant challenge that can be used to extend and enrich. Students can work individually or in small groups:

(b) Explain why there is only one possible square with the given pair of opposite vertices. Use theorems to justify your reasoning. Would this also be true if a pair of consecutive vertices were given? How many rectangles, in general, are determined if 2 vertices are given (opposite or consecutive)?

(c) Determine the coordinates of Q and S, the other pair of vertices of the square.
Note: There are many many approaches here. Students often get hung up on the distance formula leading to messy algebra (with 2 variables). There are simpler coordinate methods. You may want to provide a toolkit for students here: Graph paper, Geometer's Sketchpad, etc. Students who estimate or 'guess' the coordinates must verify (PROVE) that these vertices do in fact form a square. Students who quickly 'solve' the problem should be encouraged to find more than one method. This is the only way they will expand their thinking!

And now another coordinate problem that can be solved by a variety of methods...

In right triangle PQR, with right angle at Q, the coordinates of the vertices are:
P(-p,0)
Q(0,8)
R(r,0)
Determine the value of the area of the triangle. Assume p and r are positive.

Notes: Students should again be encouraged to try both synthetic (Euclidean) and analytical (algebraic, coordinates) methods.

General Comment: Students often forget how powerful slopes can be when solving geometry problems by coordinate methods!

Posted by Dave Marain at 4:59 PM 2 comments

Labels: coordinate problems, geometry, investigations, proof, SAT-type problems, squares