Geometry Challenge for SAT Prep or Review for Final Exam

In the coordinate plane, what is the area of ΔPQR given the coordinates P(4.5,4.5), Q(8.5,8.5), and R(6,0)?

Comments

• Example of "Grid-In" or student-constructed response question on the SAT
• This question seems more difficult than it really is. Students often give up on questions near the end of a section. DON'T!!
• Hopefully you will view the utility of questions like this as I do:
  Students become more competent problem solvers only when challenged with nonroutine problems which are not always to be found in the textbooks. Questions like these should become more routine in our texts and in our classes (not only honors!).

For further strategies, answer/solution, discussion, click Read more...


Answer: 12
Solution (no explanation, details omitted):
(1/2)(6)(8.5) - (1/2)(6)(4.5) = 12

Discussion Points

• What are some problem-solving strategies we need to review with our students here? Draw a diagram for sure but what are some other general attack strategies students should employ in triangle area problems?
• Although advanced theorems could be used here, the actual solution given above is efficient and fairly basic. But what insights are needed to use that approach? What geometry or algebra standards are being tested here?
  
• I chose this problem because coordinate geometry problems connect many important ideas in geometry and algebra. Not to mention that they are becoming more common on standardized tests like SATs, ACTs and state assessments. Besides, I enjoyed writing the question! Sometimes I'll get the germ of an idea, re-work it many times and then the question takes on a life of its own.
• If you find an error in my work or want to share your thoughts, please add a comment!
  

...Read more

Geometry Investigation: Combining Similarity, Reflection, Paper Folding in an SAT-Type Problem

The problem/investigation below lends itself to a variety of approaches:
Similar triangles
Coordinate methods
Relationships Among ⊥ Bisector, Reflection, Paper Folding (Origami), and Symmetry

Note: Parts of the problem below are appropriate for middle schoolers.

Part I
Show that the length of the ⊥ bis segment EF is 7.5.
Comments:
(1) Suggested Questions or Tips To Get Started:
(a) How does the length of EF compare to the diagonal? What would the rectangle look like if this length were equal to the diagonal?
(b) Mark (label) all known segments. Even if you cannot immediately determine a method for finding EF, what other segments can be more easily determined?
(c) Label congruent angles! This is critical!!

(2) Of course, the instructor doesn't have to give the answer away, but if the focus is on methods, you may want to consider this.

(3) Students usually struggle with recognizing similar triangles, particularly if they're not looking for them. You may want to give this as a hint after a few minutes. Even some strong honors students may be challenged. Whenever I make these kinds of claims, you can be assured I have already tried this kind of question and observed the reactions! However, coordinate methods are also powerful here and provide excellent review. It's more cumbersome than synthetic methods (Euclidean), but definitely worth discussing and is often a method of choice when other methods are not obvious. There are other methods as well using congruent right triangles, Pythagorean relationships and algebra. Then of course there are the wrong assumption methods: Assuming 3-4-5 triangles are the same as 30-60-90 triangles! As you circulate, see if anyone comes up with an answer involving √3!!

(4) For students preparing for SATs who have not had geometry for a year or more, this problem can be an excellent review even though the difficulty level is somewhat above the SATs. IMO, working a bit beyond the level of an assessment is usually the best way to prepare most students. Anyone agree?

(5) Problems that require thinking 'outside the box' and not mechanically are the most challenging for many students who view mathematics as algorithmically driven (procedures to follow).

Part II
Surely, group, you don't believe we will stop here! Generalize your result to determine an expression for the length of segment EF for an arbitrary rectangle whose dimensions are L and W. Your result will naturally be in terms of these two parameters (L and W).

Part III
Time to "reflect" on what you've done!
(a) Since EF is the ⊥ bis of diagonal AC, it follows that __ and __ are reflection images of each other.
(b) Take a large index card (ordinary filler paper is alright too but the stiffer the better). Fold the paper or card so that A and C coincide. What does this have to do with the original problem?
[Students need to "see" that the length of the fold or crease is the length of EF in the original problem!]
The mathematics of paper-folding (origami) is fascinating. The problem in this post is just an initial view of some of the underlying concepts. Students should have the opportunity to PLAY with the index card. Have them label all congruent segments and angles. Have them identify which segments or figures are reflections of each other. Have them look for a rhombus (explain why it is!), isosceles triangles, congruent right triangles, etc. Folding and unfolding the card is fascinating and instructive!

Note: Using geometry software is also instructive and should be considered as a supplement to the physical paper-folding here. There are also excellent sources for all this on the web. One of the best is the Math Forum at Drexel. An excellent investigation from the Forum can be found here. This is appropriate for both middle schoolers and high schoolers and is the kind of activity that is promoted on this blog. Check out the links at the bottom of this activity (some require subscription). It utilizes special software but can be modified. IMO, nothing replaces the need for students to hold an object in their hands.

Setting the Tone in Precalculus - Another Coordinate Investigation

Note: Read the first comment I posted which suggests a purely Euclidean geometry approach to this problem...

Don't forget our MathAnagram for Aug-Sept. Thus far we have received a couple of correct responses. You are encouraged to make a conjecture!
Look here for directions. Here is the anagram again:

PRINCE? NAH! E-ROI!

Tangent problems are usually the domain of calculus but we can keep them within the reach of geometry and algebra if we restrict our attention to circles. The calculus student spends a considerable amount of time solving a wide variety of "tangent to the curve" exercises. As any calculus instructor will tell you, many of the harder problems ask students to determine the equations of the tangent to a curve from a point not on the curve. The issue there is not the calculus. It's all about an understanding of the interface between the algebra and geometry, the essence of coordinate methods. I developed this investigation specifically to address this issue before students enter calculus. Might be another "fun" problem to start the year off with. If nothing else, it will establish the rigor of your precalculus course early on!

Part I of Investigation
Determine the coordinates of the points of tangency for the tangent lines to the unit circle from the point (0,2).
Note: Unit circle refers to the circle of radius 1, center (0,0).

The remaining parts will all refer to this same circle.

Part II of Investigation
Repeat part I for (0,3) and (0,4).
Write your observations, conjectures.

Part III of Investigation
Show that the y-coordinate of the points of tangency for the tangent lines to the unit circle from the point (0,k) is 1/k, where k ≥ 1.

Notes, Comments...
(1) The result of Part III suggests that as k increases, the y-coordinate of the point of tangency decreases (inverse ratio). Ask students what happens as k approaches 1.
Students should make sense of this visually by sketching tangent lines from various points on the y-axis above the circle.
(2) There are several effective methods for solving the above parts, however, one needs to know the fundamental relationship between a tangent line and the radius drawn to the point of tangency. From that point on, one can represent the slope in two ways or represent the y-coordinate of the point of tangent in two ways. This requires strong understanding of coordinates, graphs and algebraic relationships. You may find other methods -- share them! BTW, one could also use trig methods.
(3) I chose the unit circle and a point on the y-axis for simplicity so that the student could focus on essential ideas. However, one could generalize the result to any circle and any point outside. Have fun with that!
(4) Anyone mildly surprised by the reciprocal relationship between the y-coordinate of the point on the y-axis and the y-coordinate of the point of tangency? Can anyone make sense of that?

Starting the Year off in Precalculus/Algebra 2 - A Little Review?

Don't forget our August-September MathAnagram. No responses yet but this mathematician deserves our recognition. 'Relatively' speaking, this mathematician was truly unique and, perhaps, the last of a dying breed.

PRINCE? NAH! E-ROI!

Sometimes I found that a coordinate problem was a good way of reviewing the geometry and algebra needed to refresh memories after a 2-month layoff. Here's a fairly straightforward one that admits many different approaches and might be used to set the tone. Encourage students, working in groups, to find at least THREE different methods. This will extend the thinking of those who "solve" it rapidly and sit there complacently. This problem is a bit more appropriate for the students who completed Algebra 2 although Geometry and Algebra 1 methods are possible.
To reiterate: The problem itself is not particularly challenging. The purpose here is to provide review of several ideas, methods, theorems and strategies.

Given points A(0,0) and B(12,0). Determine the coordinates of all points C(x,y) such that ∠ACB is a right angle and ΔACB has area 18.

When Curves Collide Part II - Quadratic Systems Re-Explored!

One of MathNotations more popular posts (hundreds of views) was published one year ago this week: When Curves Collide.

Here's a variation to review the essential ideas or to use as an assessment problem or just to challenge yourself. Parts (a) thru (d) require some theoretical analysis and algebraic skill. Part (e) is the main challenge...

An Investigation for Algebra 2/Precalculus
Consider the quadratic-quadratic system:

x² + y² = 1
y = ax² -1, a>0

(a) Show that (0,-1) is always a solution to this system.

(b) For what values of the parameter 'a' will there be 3 distinct solutions to the system?
Coordinate Interpretation: For what values of 'a' will the parabola and circle intersect in 3 distinct points?

(c) For what value(s) of the parameter 'a' will two of the points of intersection be above the x-axis? Below the x-axis (in addition to (0,-1))? On the x-axis?

(d) For the case that there are 3 distinct solutions, determine the two solutions, other than (0,-1), in terms of 'a'.

(e) Now for the main problem:

Assume the graph of our system has three points of intersection: P, Q and R(0,-1). If the area of ΔPQR is 32/25, determine the coordinates of P and Q and the value of 'a'.

(f) Can you think of an even more clever variation!

Coordinate Triangle Problem - Interface between Algebra and Geometry

For Geometry or Algebra 2 students or anyone who wants a diversion...

The vertices of ΔABC are A(m,2k), B(k+11,k-2) and C(2k+6,k-2). The area of the triangle is 15.

(a) What restrictions need to be placed on k to insure there is a triangle.
(b) Given those restrictions, determine all possible values for k.

Comments:
(1) This is not intended to be a highly challenging problem. It can be used as review for a final exam, standardized tests, SATs, etc. Of course, on the SAT, the question would only ask students to grid-in one possible answer and would not generally ask about restrictions.
(2) You may want to ask your students why the value of m is irrelevant.
(3) There are two possible values for k in this problem. Challenge your students to write a revised version that would have more possibilities. Would the coordinates have to involve quadratic expressions in k?
(4) if anyone tries this in the classroom, please let us know how it went, specifically, student reaction. How was it implemented? As a warm-up, extra challenge at end of class, part of homework assignment, extra credit?

Another Quadratic Function Challenge

The line y = k intersects the graph of y = 3x² in points A and B. Points C and D are on the x-axis and ABCD is a rectangle. If the area of ABCD = 128/9, what is the value of k? Calculators not allowed. Show your method clearly!

The above question is designed for students in their second year of algebra or precalculus. It can also be used for practice for the Algebra 2 questions on the SAT, although it is somewhat above that level. Many students, including the more advanced, tend to struggle with problems like these because they don't have that much experience with coordinate geometry questions. These types of problems are critical for their later development. Once the students have done a few of these they do not find them so formidable. A useful pedagogical tool is to let them try it, review the method clearly, then erase the board and call on students to recall each step. Tell them you will do this, encouraging them to take good notes and pay careful attention! When using it on an assessment, make it a bonus the first time, then make it count. Some of my readers may recall a similar parabola problem a few months ago.

Making Connections: An Algebra Problem -- Or Is it Geometry? Or Both?

Lines L and M are perpendicular. Line L contains (0,0). Lines L and M intersect at A(5,2). Find x.

Comments:
(1) Would students in geometry be more likely to consider some or all of the following: similar triangles, altitudes on hypotenuse theorems, areas, Pythagorean approaches? If this were presented as a coordinate problem in Algebra 2, what would be the most likely approach? Would some students use the distance formula?
(2) As always, it is our role as educators to present these kinds of challenges and to encourage students to think more deeply. Making connections between algebra and geometry happens naturally for some, but certainly not for all! We must enable this dialogue via the classroomm environment we establish and the kinds of questions we ask. It is important to first decide the goal. What concepts are we trying to develop? Slope? Ratios of corresponding parts? How many students never make the connection between the two!
(3) The answer is 29/5 or 5.8. Think of at least two methods!

Two Coordinate Problems: SATs and Geometry Enrichment

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!

Going off on Tangents without Calculus!

[Update: Answers to several of these are now posted in the comments. Also, some nice discussion as well.]

To challenge Geometry, Algebra 2 and Precalculus students, we can always go back to our old friend, coordinate geometry. When I learned this way back when, it was referred to as 'Analytic Geometry'!

The following is a series of problems that review some basics of circle geometry, coordinate methods and lots of good algebra. Most of these can be found elsewhere and there are several different methods of approach. The method I'm suggesting for the first few problems is a bit different, i.e., determining the general equation of a tangent line to a circle, whose center is at the origin, at an arbitrary point on the circle. It used to be a standard formula taught in that above-mentioned course, but few students see it nowadays. Try it as in-depth investigation or exploration, starting in class or as an extension (long-term assignment or extra credit). Our AP calculus students can benefit from 'open-ended' experiences like this before they get to the AP course.

STUDENT ACTIVITY

For the first 2 questions, consider the circle whose center is at (0,0) and whose radius is 5.

1. Determine the equations of the tangent lines to this circle at the points (3,4) and (4,3). Write the equations in the form Ax+By = C. What do you notice about the results?
2. Based on the pattern of your answers in question 1, make a conjecture about the equation of the tangent line to this circle at an arbitrary point (x₁,y₁) on the circle. Now verify your conjecture 'analytically', i.e, using coordinate methods and algebra.

3. Based on the above patterns, make a conjecture about the equation of the tangent line to the circle of radius r, center (0,0) at an arbitrary point (x₁,y₁) on the circle. Verify your conjecture.

4. Now we return to the original circle of radius 5, center (0,0). Write the equations of the two tangent lines to this circle, which have a slope equal to -2. Again, write them in the form
Ax+By=C.
Note: There are many many approaches here. Discuss at least two!

5. Now, let's go outside the circle. Consider the circle of radius 1, center at (0,0) and let P have coordinates (0,2). Determine the equations of the two tangent lines to the circle through P. Also indicate the coordinates of the points of tangency.
[This 'special' case can be handled with very little algebra or computation.]

6. To generalize a bit more, consider the circle of radius r, center at (0,0) and let P have coordinates (0,2r). Again, determine the equations of the two tangent lines to the circle through P. Also, express the coordinates of the points of tangency in terms of r.

7. Final Generalization: Consider the circle of radius r, center at (0,0) and let P have coordinates (0,b) where b > r. Again, consider the 2 tangent lines to the circle, which contain P. Write an algebraic expression for the coordinates of the 2 points of tangency in terms of r and b.

The 'Power' of Geometry - Ratios of Areas

[Update: The answer, thanks to tc, and an in-depth treatment of this problem now appear in the comments. There are also some thoughts about geometry curriculum and how I develop some of these problems. I would be very interested in reader reactions to this and other problems I have written. Are they of any use for math teachers in the classroom or just curiosities to think about for the moment? Sometimes I feel that many educators just don't have the time in a packed curriculum to be able to give any of these 'enrichment' experiences. I guess I am looking for some validation here to continue writing these...]

A recently released SAT question (for copyright reasons I avoid posting exact SAT questions) motivated me to generalize the result of the problem and provide a challenge for the stronger geometry or algebra student. This problem can be solved several ways, some of which involve some 'messy' algebra. I invite our readers to find a Euclidean method that requires very little algebra and can be done mentally! When giving this type of question to our students, it is natural to want to provide hints when they become frustrated. From my own experience, I've learned to allow them to play around with it for awhile and discuss it in their groups before 'steering' them. An algebraic approach using equations of lines is certainly a worthwhile experience. The 'elegant' method I'm suggesting may not be the most desirable to show them at first. Besides, someone may devise an ingenious approach none of us would imagine if we didn't allow them to explore! Isn't that what teaching really is all about - leading the student to find her/his own path?

Ok, here's the question:
Refer to the above diagram. Lines j and n are perpendicular and contain point P(a,b) in quadrant I. Express the ratio of the area of triangle OPC to the area of triangle OPD in terms of a and b.

Another Quadratic Function Problem 2-16-07 through 2-20-07

You may want to read the comments for this post. Answers and possible solutions are discussed. There is also considerable discussion about teaching techniques for f(x-h).

The parabola problem from 2-15-07 generated some interesting discussion. I haven't had a chance to see it implemented with our Algebra 2 classes yet but I'll let you know if and when...
Today's problem is along the same lines. I'm trying to provide some problems that are exclusively high school math content for this time of year. There are dozens of outstanding problem-solving sites for MathCounts and similar middle-school competitions but there appears to be a dearth of secondary math problem-solving sites (or I haven't found them yet!). Again, how might one use the problem below? As a bonus or an extended in-class activity or a performance assessment or ??? How many would regard this question as suitable only for honors or accelerated students? My take is that if students are exposed to higher levels of thinking and know they are expected to learn how to do these and held accountable on an assessment, they will adjust. Not all will experience equal success but that's ok too! Many should be able to do part (a) or are my expectations way too high?

(a) Consider the quadratic function f(x) = 4(x+4)².
The graph intersects the line y = k, k>0, in 2 distinct points B and C.
The rectangle whose base is on the x-axis and 2 of whose vertices are B and C
has area 64. Determine the value of k. Show method clearly.

(b) Now let's generalize the result of (a).
Consider the quadratic function f(x) = a(x-h)², a>0.
The line y = k, k>0, intersects the graph of f in two distinct points B and C. The rectangle whose base is on the x-axis and two of whose vertices are B and C has area R.
(i) Explain graphically (not algebraically) why the area, R, of this rectangle is independent of h.
(ii) Express k in terms of a and R. Check that your formula for k gives the value you obtained from part (a).

Challenge Problem for 2-7-07

Day 8 - still awaiting a response from the National Math Panel...

Pls read the comments re the problems from 2-6-07. There are hints for #1 and a good discussion about #3.

Something different today. This is another one of those weekly online challenges I gave last year just before the AMC Contest. Students found it difficult but those who persisted got it. Perhaps that's the best reason to give these challenges -- to teach persistence, a quality that distinguishes some of the best researchers and problem solvers from the rest. Remember to click on the image to magnify it if it's too small.