Question

Two sides of a triangle are 18 and 11. What is the range of possible values for the third side?

Answer 1

From just above 7 to just below 29

Explanation:

For this question what we want to use is the law of cosines, also known as the cosine formula or cosine rule. It states that the square of the side opposite an angle is equal to the sum of the squares of both adjacent sides minus 2 times each side and the cosine of the angle. Like this:

`c^2=a^2+b+2-2*a*b*cos(gamma)`

Since the sum of the internal angles of a triangle has to be `180^0` we can input both sides that we know (a and b), plus the `gamma` angle into that equation with an impossibly high (180) and an impossibly low (0) value for `gamma` to get a range of values for c.

So for a low value of c (`gamma =0`), we get:

`c^2=18^2+11^2-2*18*11*cos(0)`

`c^2=324+121-396*1`

`c^2=49`

`c=sqrt(49)`

`color(red)(c=7)` (you can see that if you plug in 0.001 for the angle you will get 7.000x for the answer)

Now for a high value of c (`gamma=180`), we get:

`c^2=18^2+11^2-2*18*11*cos(180)`

`c^2=324+121-396*(-1)`

`c^2=841`

`c=sqrt(841)`

`color(red)(c=29)` (same here, if you plug in 179.999 you will get 28.999x)

So for all possible angles of `gamma` the third side can be as low as just above 7 and as high as just shy of 29