I put this question on the board the other day as I found it interesting. The idea is students find the area of the triangle. It looks like a right angled triangle, and sure enough a lot of students assumed that it was.

One or two tested it using Pythag, and found it wasn’t. Those who tested it, applied the cosine rule to find an angle, and found the area using 1/2abSinC.

Those who were less meticulous, jumped straight into 1/2 base x height.

Interestingly, the answer sheet lists it as this:

However, this is taken from a “Using Sine to find Area” worksheet, so students would likely have used the context to not use 1/2 base x height.

If I were less experienced however, I might be inclined to let those assuming this was a right angled triangle off the hook, as that gives an answer of 85.

The other way gives a decimal that rounds to 85. Don’t just rely on an answer sheet!

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When you refer to those using the cosine rule, are you referring to the Law of Cosines? Since the figure is not a right triangle, you shouldn’t use the inverse cosine because the ratio would not be accurate (unless you’re only looking for an estimation of its area). Another thought though: you had students who assumed there was a right angle (which there isn’t); what if they responded by saying that in order to calculate the area of a triangle, one must use a side of a triangle with a height perpendicular to that given side…since there isn’t such a relationship given in the triangle, I cannot calculate its area? In other words, they knew what relationship was needed (and expressed their “limitations” succinctly) but didn’t extend their thinking to consider breaking it up into two right triangles by drawing in an altitude.