kitchen table math, the sequel: help desk - triangle

## Sunday, April 10, 2011

### help desk - triangle

In triangle PQR, PQ = 4, QR = 3, PR = 6, and the measure of angle PQR is x°. Which of the folllowing must be true about x?

(A)  45 ≤ x < 60
(B)  x = 60
(C)  60 < x < 90
(D)  x = 90
(E)  x > 90

Daniel Ethier said...

(E) x > 90

If PR were equal to 5, you would have a right triangle with the Pythagorean triple 3, 4, 5. Since PR is longer than it "should" be, the angle must be more than a right angle.

Catherine Johnson said...

Thank you!

I had another 'implicit learning' moment with this problem. I was convinced that the answer had to be E, but I didn't know why until I read the explanation---- Glen said...

This is a great example of how "underlying concepts" are so heavily based on background knowledge. My guess is, almost everyone who solves it sees, "3, 4,..." and thinks, "5," then gets surprised by a 6 instead, leading them to conclude that "the 90-degree angle" is spread wider than 90 degrees.

If you don't have the "concept" of 3-4-5 to recognize, you can try scanning the supplied answers. If they mention "90 degrees," you recognize the concept that it's probably a Pythagorean Theorem problem. And if the problem has neither feature to recognize, you recognize that general questions about triangle side lengths and angle measures are usually Law of Sines or Law of Cosines problems if you're taking a math test and general trigonometry problems if taking a science or engineering test.

It's recognition all the way down.

GS said...

I drew a diagram to answer this one, and liked the question sufficiently to record it for future use!

Catherine Johnson said...

My guess is, almost everyone who solves it sees, "3, 4,..." and thinks, "5," then gets surprised by a 6 instead, leading them to conclude that "the 90-degree angle" is spread wider than 90 degrees.

That's the solution a person who's semi-fluent in h.s. geometry should arrive at -- but I didn't.

I didn't consciously remember 3,4,5 triangles when I looked at this problem -- and yet I felt convinced that the angle across from '6' had to be obtuse.