Question

line segments xy and zy are tangent to circle o. which kind of triangle must triangle xyz be? an equilateral triangle an isosceles triangle a scalene triangle a right triangle

Explanation:

Step1: Use the tangent - segment theorem

The tangent - segment theorem states that if two tangent segments are drawn to a circle from the same external point, then the two tangent segments are congruent.
In this case, \(XY\) and \(ZY\) are tangent segments to circle \(O\) from the external point \(Y\). So, \(XY = ZY\).

Step2: Recall the definition of triangle types

An equilateral triangle has \(AB=BC = CA\). An isosceles triangle has at least two sides equal. A scalene triangle has \(AB
eq BC
eq CA\). A right - triangle has one right angle.
Since \(XY = ZY\), by the definition of an isosceles triangle (a triangle with at least two congruent sides), \(\triangle XYZ\) is an isosceles triangle.

Answer:

an isosceles triangle