If a line is perpendicular to a radius at its endpoint on a circle, then the line is tangent to the circle.
Therefore, to determine whether a tangent is shown on the diagram, it is enough to determine if the line that appears to be a tangent forms a right angle with the radius.
To do so, we will use the Converse of the Pythagorean Theorem. We will substitute the side lengths into the Pythagorean Theorem.
a^2+b^2=c^2
⇓
XY^2+YZ^2? =XZ^2
Next, we will substitute the given lengths in the above equation. If we obtain a true statement, the triangle is a right triangle. If we have a right triangle, the line is a tangent. If we do not have a right triangle, the line shown is not a tangent.
Since we obtained a true statement, the triangle in the diagram is a right triangle. Therefore, XY forms a right angle with the radius. This means that it is a tangent of the given circle.
