A tangent has two defining properties
- A tangent touches a circle in exactly one place
- The tangent intersects the circle's radius at a 90° angle
Since a tangent only touches the
circle at exactly one and only one point, that point must be
perpendicular to a radius.
To test out the interconnected relationship of these two defining traits of a tangent, try the interactive exercise below. It's only when the line is tangent to the circle that the radius will hit that line at exactly one point and at this point the line segment or 'tangent' must intersect with the radius at a 90° angle.
The point where the tangent and the circle intersect is called the tangent of a circle.
In the examples below, find the line segment that is Tangent to each circle. (This line is called the tangent line) Click the button to see if you identified the right line!
Identify Tangent
AB
is tangent to the circle since the segment touches the circle once and intersects with the radius at
a 90° angle.
Identify Tangent
VK is tangent to the circle since the segment touches the circle once and intersects with the radius at a 90° angle.
What must be the length of
LM for this segment to be tangent line of the
circle with center N?
answer
For segment LM to be a tangent it intersect the radius MN at 90°. Therefore
triangle LMN would have to be a right triangle and the pythagorean theorem provides the necessary length for LM to be a tangent.
NWhat must be the length of
LM for this line to be a tangent line of the circle with center N?
answer
What must be the length of YK for this line to be tangent to the circle with center X?
answer
How many, if any, of the circles above have tangent line? In both cases X is the center of the respective circles.
Answer
Remember: that a defining property of a tangent line is that it intersects the radius of the circle at 90°. If either segments AB or XY are a tangent of the respective
circle, then these segments create right angles and right triangles. Therefore, the pythagorean theorem would have to be true for the triangles in question.
However, in both cases the pythagorean theorem is not true. Therefore the triangles are not right, and the lines in question are not tangent to the circle.
Neither circle has a tangent line
In circle 1 : 72²+ 21² ≠ 150².
In circle 2 :does not have a tangent because 20 ²+ 49 ² ≠ 50²
