Question

Line segments XY and ZY are tangent to circle O.

Which kind of triangle must triangle XYZ be?

A) an equilateral triangle
B) an isosceles triangle
C) a scalene triangle
D) a right triangle

Answer 1

It would be an Isosceles triangle

Answer 2

Combine points X and O to get segment XO and combine points Z and O to get segment ZO. Segments XO and ZO are radii of the given circle, then they are congruent. Thus, the triangle XOZ is isosceles, that gives you
`\angle OXZ\cong \angle OZX`.

Since segments XY and YZ are tangent to the circle, then
`m\angle YXO=m\angle YZO=90^(\circ)`.

Consider angles ∠ZXY and ∠XZY:

`m\angle ZXY=m\angle OXY-m\angle OXZ,\\m\angle XZY=m\angle OZY-m\angle OZX`.

Taking into account that

`m\angle OXY=m\angle OZY,\\ m\angle OXZ=m\angle OZX`,

you have

`m\angle ZXY=m\angle XZY`.

If twoo angles adjacent to the side are congruent, then this side is a base of isosceles triangle.

Answer: correct choice is B.