Friday, May 2, 2008

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.

(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?


jd2718 said...

I am curious about how differently my algebra 9th graders and my geometry 9th graders will approach this.

Will report back!

Florian said...

Neat little problem. It would be more
interesting if no side was parallel
to the coordinate axis'

Dave Marain said...

Definitely more interesting, Florian!
But remember, I'm suggesting problems that are accessible to both hojnors and non-honors groups. Too often, non-accelerated students do not get the opportunity to see these. Your suggestion would be a wonderful extension -- perhaps that will appear soon!

Yes, I'm curious about that. The geometry student may struggle with the algebra and vice versa, but I would guess that the algebra presents more difficulty for many. Let us know!