Tangent lines of circles
What is a tangent line to a circle?
A tangent line to a circle intersects the circle at exactly one point on its circumference.
Line ???XY??? is tangent to circle ???A??? at point ???X???. Point ???X??? is called the point of tangency. The radius drawn from the center of the circle to the point of tangency is always perpendicular to the tangent line. In the figure below, Radius ???\overline{AX}??? is perpendicular to line ???XY???.
How to use tangent lines of circles
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Determining whether or not a line is tangent to a circle
Example
The radius of circle ???C??? is ???5??? units. ???AB=6\text{ units}??? and ???DA=3\text{ units}???. Determine whether line ???AB??? is tangent to circle ???C???.
If line ???AB??? is tangent to circle ???C???, then the radius will be perpendicular to line ???AB??? and angle ???\angle CBA??? will be a right angle. If the triangle formed in the diagram is a right triangle, then the Pythagorean theorem will be satisfied for the triangle, so we want to verify the following equation.
???CB^2+AB^2=CA^2???
???\overline{CB}??? is a radius so we know ???CB=5\text{ units}???, and we know that ???AB=6\text{ units}???. We know that ???CA=CD+DA???, and that ???\overline{CD}??? is also a radius, so ???CD=5\text{ units}???. Since ???DA=3\text{ units}???, ???CA=5+3=8???. Now we can check the Pythagorean theorem.
???CB^2+AB^2=CA^2???
???5^2+6^2=8^2???
???25+36=64???
???61\ne64???
This means that angle ???\angle CBA??? is not a right angle, and line ???AB??? is therefore not tangent to circle ???C???.
Let’s do one more.
Example
Find the length of the radius of circle ???C???, given that line ???AB??? is a tangent line.
A radius drawn to point ???A??? will be perpendicular to line ???AB??? and form right triangle ???BAC???. Let’s call the length of the radius ???x???, then ???AC=x??? and ???CD=x???. Now we can use the Pythagorean Theorem to set up an equation and solve for ???x???.
???{{x}^{2}}+{{15.2}^{2}}={{(x+7.6)}^{2}}???
???{{x}^{2}}+231.04={{x}^{2}}+15.2x+57.76???
Subtract ???x^2??? and ???57.76??? from both sides and rearrange.
???15.2x=173.28???
???x=11.4???
The radius of circle ???C??? is ???11.4\text{ units}???.