Question
Line segments XY and ZY are tangent to circle O.
Circle O is shown. Triangle X Y Z has points X and Z on the circle. Lines X Y and Z Y are tangents that intersect at point Y outside of the circle.
Which kind of triangle must triangle XYZ be?
an equilateral triangle
an isosceles triangle
a scalene triangle
a right triangle
Circle O is shown. Triangle X Y Z has points X and Z on the circle. Lines X Y and Z Y are tangents that intersect at point Y outside of the circle.
Which kind of triangle must triangle XYZ be?
an equilateral triangle
an isosceles triangle
a scalene triangle
a right triangle
Answers
GPT-4o mini
Since lines XY and ZY are tangent to circle O at points X and Z, and they intersect at point Y outside the circle, we can use the properties of tangents to determine the type of triangle XYZ.
1. Tangents from a common external point (in this case, point Y) to a circle are equal in length. Thus, we have:
\[
XY = YZ
\]
2. Since two sides of triangle XYZ (XY and YZ) are equal, triangle XYZ must be isosceles.
Therefore, triangle XYZ is an **isosceles triangle**.
1. Tangents from a common external point (in this case, point Y) to a circle are equal in length. Thus, we have:
\[
XY = YZ
\]
2. Since two sides of triangle XYZ (XY and YZ) are equal, triangle XYZ must be isosceles.
Therefore, triangle XYZ is an **isosceles triangle**.