Question

In Triangle XYZ, measure of angle X = 49° , XY = 18°, and

YZ = 14°. Find the measure of angle Y.

Answer 1

There are two choices for angle Y:
`Y \approx 54.987^(\circ)` for
`XZ \approx 15.193`,
`Y \approx 27.008^(\circ)` for
`XZ \approx 8.424`.

Explanation:

There are mistakes in the statement, correct form is now described:

In triangle XYZ, measure of angle X = 49°, XY = 18 and YZ = 14. Find the measure of angle Y:

The line segment XY is opposite to angle Z and the line segment YZ is opposite to angle X. We can determine the length of the line segment XZ by the Law of Cosine:

`YZ^(2) = XZ^(2) + XY^(2) -2\cdot XY\cdot XZ \cdot \cos X` (1)

If we know that
`X = 49^(\circ)`,
`XY = 18` and
`YZ = 14`, then we have the following second order polynomial:

`14^(2) = XZ^(2) + 18^(2) - 2\cdot (18)\cdot XZ\cdot \cos 49^(\circ)`

`XZ^(2)-23.618\cdot XZ +128 = 0` (2)

By the Quadratic Formula we have the following result:

`XZ \approx 15.193\,\lor\,XZ \approx 8.424`

There are two possible triangles, we can determine the value of angle Y for each by the Law of Cosine again:

`XZ^(2) = XY^(2) + YZ^(2) - 2\cdot XY \cdot YZ \cdot \cos Y`

`\cos Y = (XY^(2)+YZ^(2)-XZ^(2))/(2\cdot XY\cdot YZ)`

`Y = \cos ^(-1)\left((XY^(2)+YZ^(2)-XZ^(2))/(2\cdot XY\cdot YZ) \right)`

1)
`XZ \approx 15.193`

`Y = \cos^(-1)\left[(18^(2)+14^(2)-15.193^(2))/(2\cdot (18)\cdot (14)) \right]`

`Y \approx 54.987^(\circ)`

2)
`XZ \approx 8.424`

`Y = \cos^(-1)\left[(18^(2)+14^(2)-8.424^(2))/(2\cdot (18)\cdot (14)) \right]`

`Y \approx 27.008^(\circ)`

There are two choices for angle Y:
`Y \approx 54.987^(\circ)` for
`XZ \approx 15.193`,
`Y \approx 27.008^(\circ)` for
`XZ \approx 8.424`.