Geometry help!?

Question

I need to clasify a triangle as acute, right or obtuse and i only know the measures of the three sides but i don't know how to do it.
One is 15, 8, 21. another one is 12, 16, 20.
How do you do it?

Thanks a lot!

Answer 1

12, 16, 20 is easy. Those measures are multiples of the familiar 3, 4, 5 Pythagorean triple so we know using a^2 + b^2 = c^2 will result in showing it to be a right triangle. (12^2 + 16^2 = 144 + 256 = 400 = 20^2 so that checks out.)

8, 15, 21 though, does not check out so easily. 8^2 + 15^2 does not equal 21^2 (64 + 225 = 289 (which is 17^2) not 441). One can say this for sure though, given that, if the 21 side were 17, it would be a right triangle (so that the angle between 8 and 15 would be 90, if opposite a 17 measure side), since a 21 measure "spreads" the angle opposite more than a 17 measure does, the angle between 8 and 15 has to be greater than 90 and therefore the triangle is obtuse.

Answer 2

The cosine rule. a^2 = b^2 + c^2 - 2bc * cos A
Square the longest side- 21^2 = 441 = 225 + 64 - 240 cos A
Simplify: -240 cos A = 152 so cos A is a negative number.

Cosines are positive for angles < 90 degrees (acute or sharp), negative for greater than 90 degrees (obtuse or blunt), and zero at 90 degrees (just Right). This triangle has an Obtuse angle.

Second example: 20^2 = 400 = 256 + 144
cos A is zero- you have a right angle.

Answer 3

12 16 20 is definitely right.

Answer 4

You use the trig ratios sine, cosine, and tangent and take the inverses. I can't really explain it on a computer, I'd need to show you in person, which I obviously can't do...