A line that touches a circle at exactly one point, called the point of tangency. At this point, the tangent line is perpendicular to the radius.
Imagine a ball sitting on a flat floor. The floor touches the ball at exactly one point—that's tangency. The floor (tangent line) is perfectly perpendicular to a line from the ball's center to the contact point (the radius). No matter how you tilt the flat surface, if it only touches at one point, it must be perpendicular to the radius there.
Showing a random 20 of 50 problems.
Example 1
medium
If a line passes through the center of a circle, can it be tangent to the circle? Yes or No.
Example 2
easy
What angle does a tangent make with the radius drawn to the point of tangency?
Example 3
easy
Is the distance from the center to a tangent line equal to the radius? Yes or No.
Example 4
easy
A tangent line touches circle O at T. If OT=6 and OP=10 for external point P on the tangent, find PT.
Example 5
easy
If a line is closer to the center than the radius, how many times does it cross the circle?
Example 6
hard
From external point P, PA is tangent to circle O at A, and PBC is a secant where PB=4 and PC=9. Find PA.PB = 4, PC = 9; find PA using power of a point
Example 7
easy
What is the measure of the angle between a tangent and the radius at the point of tangency?
Example 8
medium
From external point P, 17 units from the center of a circle of radius 8, find the length of the tangent segment from P.r = 8, OP = 17; find tangent length
Example 9
medium
A circle is inscribed in a triangle with sides 5, 12, 13. Find the radius of the inscribed circle (incircle).
Example 10
medium
Two tangent segments from external point P touch a circle at A and B, with angle APB = 50°. Find the angle between each tangent and the line PO (to the center).
Example 11
easy
Two tangent segments from a single external point have lengths x and 11. Find x.
Example 12
easy
Two tangents drawn from the same external point to a circle have what relationship?
Example 13
challenge
Why does a ball resting on a flat floor always touch at a point where the floor is perpendicular to the radius, no matter how the floor is tilted?
Example 14
easy
A radius to a tangent point is 5, and the tangent extends 12 from that point to an external point P. Find the distance from the center O to P.
Example 15
easy
A radius drawn to the point of tangency has length 7. From an external point P, OP=25. Find the tangent length PT.
Example 16
medium
In the two-tangent figure, angle APB = 50° (P external). Find the angle AOB at the center.
Example 17
hard
A line is tangent to a circle of radius 5 at point T. If the line's slope is 43, what is the slope of the radius OT?
Example 18
medium
True or False: A tangent line and a chord can meet at the point of tangency at any angle.
Example 19
hard
A circle of radius r is inscribed in a square of side 10. The circle is tangent to all four sides. Find r.
Example 20
medium
Two circles, radii 3 and 5, are externally tangent. How far apart are their centers?