DWG NO. 12.3 — Lesson 3 of 5

Tangent Lines and Their Properties

Unit 12: Circles · ~15–30 min

Objective

Use the tangent-radius and congruent-tangent-segment properties to solve for missing lengths and angles.

Tangents and secants

A line can meet a circle in zero, one, or two points. A tangent line touches the circle at exactly one point — the point of tangency — and stays outside the circle everywhere else. A secant line, by contrast, passes through the circle and intersects it at two points.

Two key tangent properties

Tangent–Radius Theorem: a tangent line is perpendicular to the radius drawn to the point of tangency. This turns most tangent problems into right-triangle problems.
Converse: if a line is perpendicular to a radius at the point where the radius meets the circle, that line is tangent to the circle.
Two-Tangent Theorem: if two tangent segments are drawn to a circle from the same external point, those two segments are congruent.

Radius OT meets the tangent line at point T, forming a right angle.

Worked Example 1 · Right-triangle tangent problem

ProblemCircle O has radius 9 cm. External point P is 15 cm from the center, and segment PT is tangent to the circle at T. Find PT.

1.Since PT is tangent at T, radius OT ⊥ PT, so triangle OTP is a right triangle with hypotenuse OP = 15 and leg OT = 9.

2.Apply the Pythagorean Theorem: OT² + PT² = OP², so 9² + PT² = 15².

3.Solve: 81 + PT² = 225, so PT² = 144, PT = 12 cm.

PT = 12 cm

Worked Example 2 · Two tangent segments

ProblemFrom external point Q, two tangent segments QR and QS touch circle O at R and S. QR = 5x − 3 and QS = 2x + 12. Find x and the common tangent length.

1.By the Two-Tangent Theorem, QR = QS: 5x − 3 = 2x + 12.

2.Solve: 3x = 15, so x = 5.

3.QR = 5(5) − 3 = 22.

x = 5; QR = QS = 22

Guided practice

1.Segment AB is tangent to circle O at B. What must the measure of ∠OBA be?
Show answer
90°. A tangent line is always perpendicular to the radius drawn to the point of tangency.
2.Circle O has radius 6 in. External point K is 10 in from O, with tangent segment KT. Find KT.
Show answer
OT ⊥ KT, so 6² + KT² = 10² → 36 + KT² = 100 → KT² = 64 → KT = 8 in.
3.From external point M, tangent segments MN and MP are drawn to circle O, with MN = 14. What is MP?
Show answer
MP = 14. Two tangent segments from the same external point are always congruent.
4.Tangent segments from point J measure JK = 4x + 1 and JL = 6x − 9. Solve for x.
Show answer
Set them equal: 4x + 1 = 6x − 9 → 10 = 2x → x = 5.
5.A line is drawn perpendicular to a circle's radius, but at a point 2 units past where the radius meets the circle. Is that line tangent to the circle? Explain using the Converse of the Tangent–Radius Theorem.
Show answer
No. The converse requires the perpendicularity to occur exactly at the point where the radius meets the circle. A line perpendicular to the radius at a different point either misses the circle or cuts through it as a secant, so it isn't guaranteed to be tangent.

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