Use tangent properties (tangent is perpendicular to radius at point of tangency) when relevant. | Circles | SAT Math

A tangent meets a circle once, and the radius to that point is perpendicular. That right angle is the key setup.

Core Idea

A tangent line touches the circle at exactly one point, and the radius to that point is perpendicular to the tangent. This creates a right angle you can use in a right triangle.

Understanding

Rule: A tangent touches the circle once, and the radius to that point is perpendicular.

Tangent + radius gives a right angle.
That right angle usually creates a Pythagorean theorem setup.
Two tangents from the same external point are equal.

Look for the 90° angle first; that is the whole setup.

Step by Step

Identify the tangent line and the point of tangency.
Draw the radius to the point of tangency — this radius is perpendicular to the tangent.
Form the right triangle: tangent segment, radius, and the line from center to external point.
Apply the Pythagorean theorem or trig ratios to solve.

Misconceptions

Forgetting that the tangent-radius angle is exactly 90° — this is the key to setting up the right triangle.
Assuming a secant (a line that crosses the circle at two points) has the same property — it doesn't.
Not recognizing that two tangents from the same external point are equal.

Question

Worked Example

A tangent segment from an external point $𝑃$ to a circle touches the circle at point $𝑇$ . The radius of the circle is 5, and $𝑃 𝑇 = 12$ . What is the distance from $𝑃$ to the center of the circle?

Select an answer to see the explanation