Question

How many distinct triangles can be formed for which m∠X = 51°, x = 5, and y = 2?

zero
one
two

Answer 1

One

Explanation:

With two known sides and a non-included angle, we have to consider the ambiguous case using the Law of Sines:

`\displaystyle (\sin(X))/(x)=(\sin(Y))/(y)=(\sin(Z))/(z)\\\\(\sin(51))/(5)=(\sin(Y))/(2)\\\\(2\sin(51))/(5)=\sin(Y)\\\\\sin^(-1)\biggr((2\sin(51))/(5)\biggr)=Y\\\\Y\approx18.11^\circ`

Since
`180^\circ-18.11^\circ=160.99^\circ` and
`160.99^\circ+51^\circ=211.99^\circ > 180^\circ`, then
`160.99^\circ` is not a valid measurement for the second angle.

Therefore, since there is only one possible value for the second angle, then there is one distinct triangle.