Algebra and Geometry: ReUnited in Marriage?
The MathLint for this week discusses the failed marriage between algebra and geometry, while the MathQuote offers a sinister reason for the split.
The following problems ask you to explore the possible reconcilliation of the two...with each experience handpicked to show the power of both algebra and geometry...separately and together.
Your Task: For each Conjecture or Problem, prove or solve it twice, once using algebra and once using geometry. Are you up to it?
Conjecture 1: In any quadrilateral, the lines joining the midpoints of opposite sides bisect each other.
Conjecture 2: In a parallelogram, if the diagonals are congruent, then the parallelogram is a rectangle.
Conjecture 3: In any quadrilateral, the lines joining the midpoints of adjacent sides form a parallelogram.
Conjecture 4: Any angle inscribed in a semicircle is a right angle.
Problem 1: Two cities C_{1} and C_{2} are 100 miles apart. One automobile starts from C_{1} at a certain time and travels toward C_{2} at the constant rate of 30 mph. A second automobile leaves C_{2} an hour after the departure of the first from C_{1} and travels toward C_{1} at the constant rate of 45 mph. when, and how far from C_{1}, will the two automobiles meet?
Problem 2: Two men, A and B, can do a job in 4 days, working together. At the end of 2 1/2 days, however, B quits the job, whereupon A works alone and finishes it in 5 days. How long would it take A to accomplish the whole job with no help from B?
Source: Adapted from G. Merriman's To Discover Mathematics, 1942, pp. 161184
Hint: For each conjectures, a good place to begin is to explore it using Geometers SketchPad.
For the conjectures, key needs are to label all relevant points so that you can invoke both algebraic and geometric ideas.
For the problems, the unusual task is to somehow represent them geometrically. Think "graph" using linear equations....
Solution Commentary: I refuse to spoil your opportunity to "rewed" algebra and geometry. But, if you need help, consider Merriman's text, pp. 161184, where he gives a detailed analysis of each approach, algebraically and geometrically. You should be able to find an electronic copy of the text online.
