Chords, Tangents, and Radius Relationships

Objective: Use a radius-tangent right angle.

Prerequisite skills: Pythagorean theorem and circle geometry. A radius to a point of tangency meets the tangent line at 90°. This creates a right triangle when an external point connects to the center and tangency point. The theorem depends on tangency at that exact point. Common error: applying the right-angle rule to a secant that crosses the circle twice. Restrictions and interpretation: A tangent is perpendicular to the radius at the tangency point; chord bisectors through the center are perpendicular.

Objective: Use half a chord with a perpendicular center distance.

A perpendicular from the center to a chord bisects the chord. The radius to a chord endpoint, half the chord, and the center-to-chord distance form a right triangle. Do not use the whole chord as a triangle leg. Common error: substituting the whole chord length instead of half. Restrictions and interpretation: A tangent is perpendicular to the radius at the tangency point; chord bisectors through the center are perpendicular.

Objective: Use a right triangle from center, tangency point, and external point.

For external point P and tangent point T, OT⊥PT, so OP is the hypotenuse of △OPT. Tangents drawn from the same external point also have equal lengths. Apply the geometry before doing arithmetic. Common error: adding the radius square instead of subtracting it from the hypotenuse square. Restrictions and interpretation: A tangent is perpendicular to the radius at the tangency point; chord bisectors through the center are perpendicular. Desmos and method choice: Use Desmos for arithmetic or a graph/table check when that reduces work. Enter parentheses explicitly and verify the requested quantity and units. Use the algebraic steps shown here for an exact result; a rounded graph display is not proof of exact equality.