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
The Correct Answer and Explanation is:
The correct answer is an isosceles triangle.
Here’s an explanation:
In the given scenario, triangle XYZ has two line segments, XY and ZY, that are tangent to the same circle O. A key property of tangent lines to a circle is that they are equal in length from the point of tangency to the point where the tangent intersects the line connecting the center of the circle (O) to the point of tangency. This is called the tangent-secant theorem. In this case, because XY and ZY are tangents to the circle from the same external point Y, the lengths of these two segments must be equal, i.e., XY = ZY.
Now, consider triangle XYZ:
- Since XY = ZY (as the lengths of the tangents from a common external point to a circle are equal), triangle XYZ has two sides of equal length: XY and ZY.
- A triangle with two equal sides is classified as an isosceles triangle.
The property of the tangents also means that angle XOY (where O is the center of the circle and OX and OY are radii) is equal to angle ZOY because these angles are formed by radii from the same point O. Therefore, the angles at X and Z formed by the tangents are also equal, reinforcing the isosceles nature of the triangle.
Since the triangle has two equal sides, it cannot be scalene (where no sides are equal), equilateral (where all sides are equal), or a right triangle (which would require a 90-degree angle between two sides). Hence, triangle XYZ must be an isosceles triangle.
In summary, the fact that XY = ZY makes triangle XYZ an isosceles triangle, which is the correct answer.