Question

How would I solve this?

Answer 1

9514 1404 393

Answer:

  not tangent

Step-by-step explanation:

The triangle has side lengths 5, 12, 14. You can compute the "form factor" for the triangle:

  a^2 +b^2 -c^2 . . . . . . . . where 'c' is the longest side

  = 5^2 +12^2 -14^2

  = 25 +144 -196 = -27

The negative value tells you this triangle is obtuse. If AB were a tangent, it would be perpendicular to the radius—the triangle would be a right triangle.

If you compare this calculation to the Pythagorean theorem, you see that the length AB is longer than the length √(25+144) = 13 that is necessary for the triangle to be a right triangle. That means the angle at A is greater than 90°.

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Additional comment

This "form factor" calculation is part of the calculation you would do using the Law of Cosines to determine the largest angle. The sign of the "form factor" tells you the sign of the cosine of the angle. Angles whose cosine is negative are greater than 90°. A positive "form factor" indicates an acute triangle.

I find computing the "form factor" in this way makes interpretation of the result fairly easy. For me, it eliminates the confusion I had when the numbers of the Pythagorean theorem didn't add up. For me, it was too much work to figure whether the triangle was acute or obtuse, and I often got it wrong.

For those interested, the angle measure is arccos((a^2+b^2-c^2)/(2ab)). Here, that's arccos(-27/(2·5·12)) ≈ 103.003°.