Twitter Problem 10-18-14
If (a,b),(-a,-b) are opposite vertices of a square, show that its area=2(a^2+b^2)
EXTENSION: What if (a,b),(-a,-b) are adjacent?
(1) What do you believe will challenge your geometry students here? The abstraction? "Show that"?
(2) Predict how many of your students would "complete the rectangle" by incorrectly drawing sides || to the axes?
(3) Even if not an assessment question, is it a good strategy to "plug in" values for a&b? This is worthy if more dialog IMO...
(4) How many of your students would question the lack of restrictions on a&b? Would most place (a,b) in 1st quadrant without thinking? So why doesn't it matter!
(5) Is it worth asking students to learn the formula "one-half diagonal squared" for the area of a square? I generally don't promote a lot of memorization but this one is useful!
Answer to extra question: 4(a^2+b^2).
Ask your students to explain visually why this area is TWICE the area of the original square!