Wednesday, July 11, 2007

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!