The Power of Visualization: Another View of Inscribed vs. Circumscribed

A student, Melissa, in my SAT class, shared her way of thinking about a well-known problem that appeared on this blog awhile back as part of a larger investigation. You may recall the post about the ratio of the areas of circumscribed and inscribed figures.

I told her I would celebrate her method on my blog, which would definitely be viewed by less than or equal to a million people a day! One of my students asked me why I don't simply write a book about all these ideas and I replied, "I'm thinking about it." He asked what the title would be and I replied, "What I've Learned From My Students."

Here's the problem/investigation/challenge/activity///// for you or your geometry students:

Consider the diagram above. Assume that it depicts two squares. The smaller inscribed square is formed by joining the midpoints of the larger square.

(1) Explain why the area of the inscribed square is one-half of the area of the larger.
Easy so far...

(2) Consider the circle circumscribed about the smaller square, i.e., it passes through its 4 vertices. Explain why this circle is inscribed in the larger square. This requires that you show that the 4 sides of the larger square are tangent to the circle. [By the way, most students would assume this is obvious from the diagram, but ...]

Not impressed? Deja vu all over again as Yogi would say? This is how Melissa demonstrated that, for a given circle, the area of the circumscribed square is twice the area of the inscribed square. Is this the method you have seen or used yourself? Consider that she used no variables, didn't plug in particular values for the dimensions, etc. She drew the diagram and basically said that the diagram proves itself! A proof without words, so to speak (that might even make Sidney Kung proud). I congratulated this young lady and the class applauded. Interesting how students immediately recognize someone's brilliance...

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.

Challenging Geometry: Circles Inscribed in Quadrilaterals, Right Triangles

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Update²: See the awesome article in MathWorld on tangential quadrilaterals for more info re problem #2.
Update: See Comments section for some answers, solutions.

Part (a) of each of the following are somewhat difficult questions that can be found in some geometry textbooks. These are numerical exercises and good practice for the more difficult SAT-types of questions or for math contests. The last part of each question is an extension or generalization of the problem. Texts do not often ask students to delve beneath the surface and look for general relationships.

1. A circle of radius 4 is inscribed in a right triangle with hypotenuse 20.
(a) Find the perimeter of the triangle without using the Pythagorean Theorem. Justify your reasoning.
(b) Using the Pythagorean Theorem, show that the triangle is similar to a 3-4-5 triangle.
Note: Many students tend to guess multiples of 3-4-5 when doing these. Sometimes they get lucky but they need to prove it!(c) PROVE in general that the perimeter of a right triangle is twice the sum of its hypotenuse and the radius of its inscribed circle. Again, no Pythagorean Thm allowed.
Note: There are well-established formulas for the inradius of a triangle. Our objective here is to look at one special case.
2. A circle is inscribed in a quadrilateral which has a pair of opposite sides equal to 12 and 18. Neither pair of opposite sides of the quadrilateral is parallel.
(a) Find the perimeter of the quadrilateral. Justify your reasoning.
(b) PROVE in general that the perimeter of a quadrilateral in which a circle is inscribed equals twice the sum of either pair of opposite sides.
Note:: Not all quadrilaterals have an inscribed circle, so this is a strong condition.

Note: As always, these results need independent verification. I welcome your comments and edits!