Graduate Management Admission Test (GMAT) Practice Test

What relation does a line that intersects a circle at exactly one point have to the radius?

1. It is parallel to the radius

2. It is collinear with the radius

3. It is tangent and perpendicular to the radius

4. It is a chord of the circle

The correct answer is: It is tangent and perpendicular to the radius

When a line intersects a circle at exactly one point, that point of intersection is known as a tangent point. A tangent line, by definition, touches the circle at only this single point and does not enter into the interior of the circle. The key relationship between the tangent line and the radius extends from the fact that at the point of tangency, the radius drawn to that point is perpendicular to the tangent line. This means that if you were to draw a line segment from the center of the circle to the tangent point, it would form a right angle with the tangent line. This principle is rooted in the geometric definitions of tangents and is fundamental to circle properties. Thus, the correct choice accurately states that the tangent is perpendicular to the radius at the point where they meet. This demonstrates not only the nature of tangents in relation to circles but also emphasizes the importance of understanding geometric relationships in the GMAT context.