Tangent to a Circle
In short: A tangent to a circle is a line that kisses the circle at a single point and meets the radius drawn to that point at a right angle.
GeometryGeometryG-C.A.4
What is Tangent to a Circle?
A tangent to a circle behaves like a ruler laid against a wheel: it meets the rim at one point, the point of tangency, and then runs off without cutting inside.
Two facts do most of the work. The radius to the point of tangency is perpendicular to the line, creating a square corner. And from a point outside, the two tangent segments reaching the circle are equal in length. With the Pythagorean theorem they give the tangent length: if the outside point lies at distance \(d\) from the center, the length is \(\sqrt{d^2 – r^2}\).
How to spot it
- A figure shows a line touching a circle at one point, with a small square mark between it and the radius, a tangent to a circle.
- The wording says tangent, touches at exactly one point or point of tangency.
- A point outside a circle sends two segments to the circle that touch it, and their lengths are compared.
- A belt, rope or wire leaves a pulley or wheel in a straight line.
The method
- Draw the radius to the point of tangency; it forms a right angle with the tangent to a circle.
- Build the right triangle: radius, tangent segment, and the segment from the center to the outside point.
- Apply the Pythagorean theorem: \(\text{tangent}^2 + r^2 = d^2\), then solve.
- If two tangent segments come from one outside point, set their lengths equal.
Tangent to a Circle: solved example
Problem
A guard cable runs from a point \(P\) that is 10 feet from the center of a round water tank of radius 6 feet, and touches the tank at \(T\) without crossing it. How long is the cable segment \(PT\)?
Solution
The cable is tangent at \(T\), so the radius \(OT\) is perpendicular to it, and triangle \(OTP\) has a right angle at \(T\).
The hypotenuse is \(OP = 10\) and one leg is \(OT = 6\). Then \(PT = \sqrt{10^2 – 6^2} = \sqrt{64} = \) 8 feet.
Try it
Question 1
A lamp post stands 13 cm from the center of a round coin of radius 5 cm on a map model. How long is the straight segment from the post that just touches the coin?
Show the answer
\(\sqrt{13^2 – 5^2} = \sqrt{144} = 12\) cm.
Question 2
Two tangent segments from one outside point measure \(3x + 2\) and \(5x – 10\). Find \(x\) and the length.
Show the answer
They are equal: \(3x + 2 = 5x – 10\), so \(x = 6\) and each length is \(3(6) + 2 = 20\).
Tangent to a Circle vs. Tangent
Here tangent names a line that brushes a circle once and meets the radius at a right angle. In trigonometry, tangent names a number: opposite divided by adjacent in a right triangle. One is drawn with a straightedge, the other computed from two legs, and they only share a name.
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