Question

which statement is false about a circles tangent line?
a. it forms a perpendicular angle with the radius at the point of tangency
b. it intersects the circle at exactly one point
c. it forms a 45 - degree angle with the radius at the point of tangency
d. it never passes through the circles center

what role does the perpendicular bisector play in constructing a tangent to a circle from an external point?
a. it is parallel to the tangent lines.
b. it helps us find the midpoint of the line connecting the center of the circle to the external point. the midpoint becomes the center of a new circle, and the points where the new circle intersects the original circle are the points of tangency.
c. it intersects the circle at only one point.
d. it passes through the center of the circle but does not intersect the circle.

which of the following is not a property of a tangent line to a circle?
a. it passes through the center of the circle
b. the point of tangency is the only point of intersection with the circle
c. it is perpendicular to the radius at the point of tangency
d. it touches the circle at exactly one point

Explanation:

Step1: Recall tangent - radius property

A tangent to a circle forms a 90 - degree (perpendicular) angle with the radius at the point of tangency, not 45 - degrees. It intersects the circle at exactly one point and never passes through the circle's center. So for the first question, the false statement is the one about the 45 - degree angle.

Step2: Understand perpendicular bisector role

When constructing a tangent to a circle from an external point, we find the mid - point of the line connecting the center of the circle to the external point using the perpendicular bisector. This mid - point is the center of a new circle, and the intersection points of the new circle and the original circle are the points of tangency.

Step3: Recall tangent properties

A tangent to a circle does not pass through the center of the circle. It touches the circle at exactly one point (the point of tangency) and is perpendicular to the radius at that point.

Answer:

1. C. It forms a 45 - degree angle with the radius at the point of tangency
2. B. It helps us find the midpoint of the line connecting the center of the circle to the external point. The midpoint becomes the center of a new circle, and the points where the new circle intersects the original circle are the points of tangency.
3. A. It passes through the center of the circle