Saturday, December 8, 2007
The Kite Problem Revisited -A View From the 'Exterior'
A post from 7-17-07 generated some wonderful comments and solutions. I decided to bring it back using a slightly different diagram. This time I'm asking for a solution path using exterior angles, a tool students often overlook. The most efficient approach may still be the one tc used, involving the sum of the interior angles of a quadrilateral, but the challenge here is to find another way...
Heres' the problem:
Assume Q, S and T are collinear. Determine the value of a+b+c.
Note: Again, there are many wonderful approaches here. Try to use the suggested one...
Posted by Dave Marain at 8:01 AM 11 comments
Labels: 3-4-5 triangles, geometry, quadrilaterals, SAT-type problems
Sunday, November 18, 2007
Circles, Chords, Tangents, Similar Triangles and that Ubiquitous 3-4-5 Triangle
[As always, don't forget to give proper attribution when using the following in the classroom or elsewhere as indicated in the sidebar]
The cone in the sphere problem led me to an interesting relationship in the corresponding 2-dimensional case with a surprise ending. (Only a math person would compare a math problem to a mystery novel!). The following investigation allows the student to explore a myriad of possibilities: from similar triangles to the altitude on hypotenuse theorems to Pythagorean, to chord-chord or secant-tangent power theorems, coordinate methods, draw the radius technique, etc. Sounds like this one problem might review over 50% of a geometry course? You decide for yourself! Just remember -- one person is not likely to think of every method. Open this up to student discovery and watch miracles unfold...
STUDENT ACTIVITY OR READER CHALLENGE
In the diagram above, segment AF is a diameter of the circle whose center is O, BC is a tangent segment (F is the point of tangency), BC = AF and BF = FC. Segments AB and AC intersect the circle at D and E, respectively. Lots of given there! Perhaps some unnecessary information?
(a) If AF = 40, show that DE = 32.
Notes: To encourage depth of reasoning, consider requiring teams of students to find at least two methods.
(b) Let's generalize (of course!). This time no numerical values are given. Everything else is the same. Prove, in general, that DE/BC = 4/5.
(c) So where's the 3-4-5 triangle (one similar to it, that is)? Find it and prove that it is indeed similar to a 3-4-5.
Posted by Dave Marain at 6:43 AM 3 comments
Labels: 3-4-5 triangles, circles, geometry, investigations, similar triangles, tangents
Tuesday, July 24, 2007
Are all triangles in the universe similar to 3-4-5?
(a) AB = 8, BC = 6 in rectangle ABCD. Find the lengths of all segments shown in the diagram above.
Specifically: BD, CF,DE, BE, FE, CE
Comments: This is another in a series of rectangle investigations. To deepen student understanding of triangle relationships and to provide considerable practice with these ideas, the question asks for more than just one result. Students should be encouraged to first draw ALL of the triangles in the diagram separately and recognize why they are all similar! Using ratios, students should be able to find all the segments efficiently. One could also demonstrate the altitude on hypotenuse theorems as well!
(b) In case, students need a bit more of a challenge, have them derive expressions for all of the above segments given that AB = b and BC = a. To make life easier, assume b> a. This should keep your stronger students rather busy! This algebraic connection is powerful stuff. We want our students to appreciate that algebra is the language of generalization.
Posted by Dave Marain at 7:45 AM 5 comments
Labels: 3-4-5 triangles, altitude on hypotenuse, generalization, geometry, right triangles, similar triangles