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Can these sides form a triangle?

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Ok so I'm stuck on these 4 geometry problems. I can't ask my teacher for help and the internet has been useless so far.

Tell if these measures can be the side lengths of a triangle. If so, is the triangle acute, right, or obtuse. Explain how you know.

1. (9,12, and 16)     2. (11,14, and 27) 3. (1.5,3.6, and 3.9) 4. (2,3.7, and 4.1)

Feb 14, 2016

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1.   [9, 12, 16]       the triangle is possible

The triangle   9-12-15  is a right triangle with the angle opposite the longest side = 90°

Therefore.......the angle opposite the longest side here will be > 90°  since 16 >15

Then, this trangle is obtuse  [we can prove this using the Law of Cosines]

2. (11,14, and 27)   this triangle is not possible

Remember that the sum of any two sides of a triangle must be greater than the remaining sde.....but....     11 +  14   = 25         and 25 <  27       ....so......no triangle is formed

3. (1.5,3.6, and 3.9)

Remember that, if we multiply every side by 10, we will have a similar triangle to the one given....so.....this triangle will be    15 - 36 - 39

Note   15^2  + 36^2  = 39^2   ....so    this triangle is a right triangle

4. (2,3.7, and 4.1)      multiply by 10 again and we have    20 - 37 - 41

Note that   sqrt[20^ + 37^2]   =  about 42.05

So.....since  a right triangle is formed  with sides of 20, 37 and   ≈ 42.05

And the angle opposite the right triangle with a side of about 42.05 would be 90°

But this triangle has a third side which  is < 42.05

Thus.....the angle opposite the side 41 in a 20 - 37 - 41 triangle will be < 90°

So.....this triangle is acute because the other two angles will also be acute [we can prove this using the Law of Sines]   Feb 14, 2016
edited by CPhill  Feb 15, 2016