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

A Geometry Tribute to Cinco de Mayo

Correction: Jonathan pointed out that I did not specify the order of the vertices. Thanks, Jonathan! Here is the revised version in which A and C are opposite vertices as are B and D:

Consider parallelogram ABCD, three of whose vertices are A(0,0), B(2,3) and D(3,2).

Find the coordinates of C and the area.

Of course, we expect our Geometry students to celebrate even more by generalizing:


Note: This has been revised for the reasons stated in the correction at the top.

Three of the vertices of a parallelogram ABCD are A(0,0), B(a,b) and D(b,a), where b>a>0.

(a) Show that vertex C has coordinates (b+a,b+a).

(b) Prove that this figure is actually a rhombus.

(c) Show that its area is b² - a². Can you find FIVE ways? (ok, that's a stretch but anything is possible on May 5th!).

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?

Classic AMC Contest Square Dissection Problem and more...

Recognize this diagram from a famous math contest problem which I first saw many years ago on an old AHSME contest (now known as AMC)? We'll start with this one and then modify it, creating variations on the basic theme. Finally, we will ask our readers/students to generalize the result algebraically.


In the diagram at the left, nothing is labeled, so we will describe it verbally and hope it will make sense.
We start with a square and dissect it by drawing 4 segments, each connecting a vertex to a midpoint of a side.

THE CLASSIC
Explain why the area of the shaded region is one-fifth of the area of the original square.

Notes/Comments:
(1) This is a wonderful exercise to develop spatial reasoning and to demonstrate a visual approach to a geometry problem when dimensions are not given. Of course, one could use an algebraic or numerical approach if one chooses.
(2) Students who 'see' the jigsaw puzzle approach of rearranging the pieces rarely consider what assumptions are being made. To make the problem even more meaningful, the instructor could ask why the shaded region is, in fact, a square.
(3) Simpler versions of this often appear on the SATs.

VARIATION #1

This time, both diagonals are drawn. The additional two segments join the midpoint of the bottom side to the midpoints of two other sides.

(a) The red shaded region (does it have to be a square?) is not one-fifth of the original square. What fractional part is it?
(b) The total shaded area is what part of the original square?

VARIATION #2

This time the smaller segments divide the sides into a 1:2 ratio. The figure is not drawn to scale. The 3 segments on the base are supposed to be equal!

(a) The blue shaded region (is it a square?) is now what fractional part of the original square?
(b) The total shaded area is now what part of the original square?



THE GENERALIZATION OF VARIATION 2
Use the diagram from Variation 2. Assume the original square has a side length of 1 unit. If the smaller segments divide the sides of the square into an x:(1-x) ratio, do parts (a) and (b) again, expressing your results in terms of x. What restrictions on x make sense here? Make sure your expressions agree with the results above.

Inside or Outside? A Geometry Investigation...

[A special thanks to Mike for correcting the error I made in question 3 below. It has now been updated!]

BACKGROUND
Students in geometry often see problems involving inscribed and circumscribed circles, squares, triangles and other polygons. Questions involving such diagrams often appear on standardized tests and on math contests as well. Although this particular post focuses only on ratios of areas involving squares, equilateral triangles and circles, I am planning a series of investigations which delves far more deeply, requiring students to discover and verify more general relationships for polygons of n sides. Further, as the number of sides of the polygons increase, students will be asked to analyze the ratio of the areas of the circumscribed to the inscribed polygons and consider if they approach a limiting value. Thus this series of activities prepares students for the calculus as well. Students are also introduced to the duality principle, useful for later study in advanced geometry. A strong background in geometry is needed and, at some point, a knowledge of trigonometric ratios is required. This could be a culminating project for a marking period or the year. I hope you enjoy this and will save it for the school year or ...

Note: Although it would appear to be more logical to have students determine ratios for the triangles first (n = 3), the diagram shows the squares (n=4) on top because the analysis is somewhat easier.

Note: Although this is couched as an investigation for the classroom, my readers are invited to attempt the questions below and suggest various approaches to finding the ratios. Those experienced in this topic will find each question fairly straightforward, however, consider the bigger picture here! Those whose geometry is rusty will need to review some basic properties of circles and polygons.



FOR THE STUDENT:
1. The diagram at the upper left depicts a circle circumscribed about a square and a second circle inscribed in the square. Verify that the ratio of the area of the inscribed circle to the area of the circumscribed circle is 1:2.
2. The diagram at the upper right may be thought of as the dual to the first diagram, in that each circle has been replaced by a square and the square by a circle. Show that the ratio of the area of the inscribed figure to the area of the circumscribed figure is invariant, that is, it remains 1:2.
3. The diagram at the lower left depicts a circle circumscribed about an equilateral triangle and a second circle inscribed in the triangle. Show that the ratio of the area of the inscribed circle to the area of the circumscribed circle is 1:4.
4. Similar to #2 except that the circles and the triangle have now been interchanged. Again, show that the ratio is invariant.

Two Geometry Problems - Dozens of Methods to Ponder...

ABCD is a rectangle in both figures. Figures are not drawn to scale.

1. In Figure 1, E is the midpoint of side BC. If the area of AECD is 1, what is the area of ABCD?

2. In Figure 2, E is a point on side AB such that AE:EB = 3:1. F is a point on side BC such that
BF:FC = 2:1. If the area of DEFC = 1, what is the area of ABCD?


Notes:
(a) Although the first question is very similar to an actual SAT problem, both questions admit multiple solution paths and strategies that can help students develop both their spatial reasoning and analytical skills.
(b) The first question is similar to some textbook questions and would be rated as above-average difficulty on the SAT's, although most visiting this site would not consider it difficult. There are many variations on these kinds of problems. The most famous one is to consider the figure formed by connecting the midpoints of the sides of a rectangle and showing its area is half of the whole.
(c) The 2nd question is more discriminating and requires more than intuition.
(d) How many of today's students have a well-developed sense of fractional parts of the whole?
(e) Try to imagine how a variety of learners, say 9th or 10th graders, would approach these. Do you think the majority would attempt a visual approach - cutting up each irregular shape into common figures or perhaps dissecting the entire rectangle into equal parts? Would any students consider plugging in arbitrary lengths for the sides of the original rectangles even though specific values are given (one could then use similarity to finish it)? Is it a good strategy to encourage students to assume each rectangle is in fact a square? How many students would attempt an algebraic setup?
(f) What is the point of spending so much time discussing various approaches to a problem? Is it really worth the time expended when there is so much more material that one must complete? You all know my "less is more" mentality and that there are no shortcuts to developing problem-solving facility.
(g) Which is more important for learning to take place: Allowing the student to struggle but arrive at a solution with some strategic guidance from the instructor OR Allowing dialogue to occur enabling students to see how their peers are approaching it? What kinds of questions might the instructor ask in facilitating the activity? What exactly is our role when students are engaged? Is there a course one can take or a book from which educators can learn such pedagogy or does one learn from experience and/or from their peers?
(h) Is this more about solving geometry problems or developing problem-solving skills? Will I ever stop asking such inane questions...

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.

Lattice Points Problem Part 2: Circles, Gauss' Circle Problem and Pick's Theorem

Eric Jablow inspired me to develop the following extended enrichment actvity/project for Geometry students. You can research the general solution of Gauss' Circle Problem in MathWorld but for this application it isn't necessary to use that formula.

Consider the circle of radius 5 centered at the origin.

(a) Determine the coordinates of the 12 lattice points on this circle. Recall that lattice points are points whose coordinates are both integers.
Note: Is it reasonable from symmetry arguments that the number of lattice points is divisible by 4?
(b) Using graph paper, show there are 69 lattice points INSIDE this circle. Describe your counting method.

The total number of lattice points inside or on our circle is 69+12 = 81. This agrees with the result from Gauss' formula but we will now 'approximate' this result using Pick's Theorem which gives the relationship among interior, boundary points and the area of a polygon whose vertices are lattice points.

(c) Consider the inscribed dodecagon formed by connecting the 12 lattice points from part (a). Determine the lengths of the sides of this polygon.

(d) By dividing the polygon into 12 triangles show that the area of this polygon is 74. No trigonometry, just Pythagorean and basics!
Hint: This is not a regular polygon but you can still divide it into 12 isosceles triangles.
Comment: Do you find it surprising that the area is rational (in fact, integral), considering that the sides are irrational?

(e) Pick's Theorem states that the area of a polygon whose vertices are lattice points is given by the formula A = I + B/2 - 1 where I = the number of interior lattice points and B = the number of boundary lattice points, that is, points on the polygon.
Show that Pick's Theorem leads to I = 69.

(f) For our problem the number of lattice points inside the circle matched the number of points inside the inscribed polygon. A coincidence? Whether it's true or not, explain why this result seems to make sense.

(g) To further investigate this 'coincidence', change the radius to 10.
(i) Show, by counting, that there are 317 lattice points inside or on this circle.
(ii) Show that there are still 12 lattice points on this larger circle.
(iii) Show that there are 303 lattice points inside or on the resulting dodecagon.
[As before, find the area w/o trig and use Pick's Thm. Note: To find the area of the polygon use (d) and ratios!]
(iv) Show that the point (4,9) is inside this circle but outside the dodecagon! This suggests why Pick's theorem fails in this case! Why?

Good luck! This is an extended challenge that I will leave up for several days and invite comment. Whether you implement it in a classroom or not, enjoy!

[By the way, some of the numerical results (like 303) have not been independently verified. If you find an error, let me know!]

Problems 3-5 thru 3-6-07: Geometry and Reasoning

Note: The problem below was chosen for the March 23rd Carnival of Mathematics. If you'd like to see a different kind of mathematical challenge, I recommend you also visit the posting for 3-30-07 (Challenging Geometry: Circles Inscribed in Quadrilaterals, Right Triangles).



Today's questions involve well-known ideas from geometry. Similar questions have appeared on SATs and math contests.
Some suggestions: Use it for in-class enrichment or assign it for extra credit outside of class. The first part lends itelf to a fairly simply visual approach (cutting up the square and matching the pieces), but the second is more sophisticated. Encourage the visualization but require the analytical approach as well!
Reviews: 30-60-90, equilateral, areas, symmetry, etc.