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Constructing Tangents
to Circles

Learn to construct a tangent line from an external point to a circle using compass and straightedge, and understand why the construction works.

Geometry

What Is a Tangent Line?

A tangent line to a circle is a line that touches the circle at exactly one point, called the point of tangency. Unlike a secant (which crosses through two points), a tangent just "grazes" the circle's edge.

\(O\)
\(T\) (point of tangency)
\(r\)

Why Tangent Constructions Matter

Engineering
Gear teeth contact along tangent lines: smooth power transfer depends on accurate tangent geometry.
Optics
Light reflects off a curved mirror along the tangent at the point of contact, following the law of reflection.
Architecture
Smooth transitions between arcs and straight walls require tangent-point precision in CAD drawings.

Key Vocabulary

External Point
A point P lying outside the circle, from which we draw the tangent.
Point of Tangency
The single point T where the tangent line touches the circle.
Semicircle on \(\overline{OP}\)
A circle with diameter \(\overline{OP}\) used in the construction: its intersection with the original circle gives T.
Thales' Theorem
Any angle inscribed in a semicircle is a right angle: the reason this construction works.

The Tangent-Radius Theorem

\(O\)
\(T\)
\(r\)
 tangent line
\(90^\circ\)

In symbols: \(\overline{OT} \perp \ell\) where \(\ell\) is the tangent at T. This perpendicularity is the foundation for the construction.

Thales' Theorem: The Key Insight

To force a right angle at T, we use Thales' Theorem: any angle inscribed in a semicircle is \(90^\circ\). If we build a circle with diameter \(\overline{OP}\), then every point on that circle "sees" \(\overline{OP}\) at a right angle.

\(O\)
\(P\)
\(M\)
\(T_1\)
\(T_2\)
 Auxiliary circle (diameter OP)
guarantees \(\angle OTP = 90^\circ\)

Tangent vs Secant vs Chord

Tangent Secant Chord
Intersection points Exactly 1 Exactly 2 Exactly 2
Extends beyond circle? Yes (line) Yes (line) No (segment)
Angle with radius \(90^\circ\) \(\neq 90^\circ\) Varies
From external point? Yes Yes No (endpoints on circle)

Two Tangent Segments from One Point

From any external point P, there are always exactly two tangent lines to a circle. The tangent segments \(\overline{PT_1}\) and \(\overline{PT_2}\) are congruent.

\(O\)
\(P\)
\(T_1\)
\(T_2\)

How to Construct the Tangent

1
Draw segment \(\overline{OP}\) connecting the center O to the external point P.
2
Find the midpoint M of \(\overline{OP}\) by constructing the perpendicular bisector.
3
Draw a circle centered at M with radius \(MO = MP\). This circle has diameter \(\overline{OP}\).
4
Mark the intersections \(T_1\) and \(T_2\) of this auxiliary circle with the original circle.
5
Draw \(\overrightarrow{PT_1}\) and \(\overrightarrow{PT_2}\): these are the two tangent lines.

Why It Works

1
T lies on the auxiliary circle with diameter \(\overline{OP}\), so by Thales' Theorem: \(\angle OTP = 90^\circ\).
2
T also lies on the original circle, so \(\overline{OT}\) is a radius.
3
Since \(\overline{OT} \perp \overline{PT}\) and \(\overline{OT}\) is a radius, by the converse of the Tangent-Radius Theorem, line \(\overleftrightarrow{PT}\) is tangent to the circle.

Example 1: Finding Tangent Length

A circle has center O and radius \(r = 5\). Point P is \(13\) units from O. Find the length of the tangent segment \(\overline{PT}\).

\(O\)
\(P\)
\(T\)
\(5\)
\(?\)
\(13\)

Example 2: Finding the Radius

A tangent from point P touches a circle at T. If \(PT = 24\) and \(OP = 25\), find the radius \(r\).

Example 3: Coordinate Tangent

Circle with center \(O(0,0)\) and radius \(r = 5\). External point \(P(13,0)\). Find the midpoint and the coordinates of the tangent points.

1
Midpoint of \(\overline{OP}\): \(M = \left(\frac{0+13}{2},\, 0\right) = (6.5,\, 0)\).
2
Auxiliary circle radius: \(MO = 6.5\). The auxiliary circle is \((x-6.5)^2 + y^2 = 42.25\).
3
Intersect with \(x^2+y^2=25\): Subtract to get \(x = \frac{25}{13} \approx 1.923\), then \(y = \pm\frac{60}{13} \approx \pm 4.615\).
4
Tangent points: \(T_1 = \left(\frac{25}{13},\, \frac{60}{13}\right)\) and \(T_2 = \left(\frac{25}{13},\, -\frac{60}{13}\right)\).

Verifying the Result

We can verify Example 3 by checking two conditions at each tangent point:

✓
\(T_1\) lies on the circle: \(\left(\frac{25}{13}\right)^2 + \left(\frac{60}{13}\right)^2 = \frac{625+3600}{169} = \frac{4225}{169} = 25\). ✓
✓
\(\overline{OT_1} \perp \overline{PT_1}\): \(\vec{OT_1} \cdot \vec{PT_1} = \frac{25}{13}\!\left(\frac{25}{13}-13\right) + \frac{60}{13}\!\left(\frac{60}{13}-0\right) = \frac{25(-144)+60(60)}{169} = 0\). ✓
✓
Tangent length: \(PT_1 = \sqrt{\left(13-\frac{25}{13}\right)^2+\left(\frac{60}{13}\right)^2} = \sqrt{\frac{144^2+60^2}{169}} = \sqrt{\frac{24336}{169}} = \frac{156}{13} = 12\). ✓

Common Mistakes

✕ Wrong Hypotenuse
In the right triangle \(\triangle OTP\), students use \(\overline{PT}\) as the hypotenuse instead of \(\overline{OP}\). Remember: the hypotenuse is always the side opposite the \(90^\circ\) angle, which is \(\overline{OP}\).
✕ Skipping Midpoint Step
Drawing the auxiliary circle centered at O or P instead of at the midpoint M. The circle must have diameter \(\overline{OP}\) to guarantee the right angle via Thales.
✕ Only One Tangent
Forgetting there are two tangent lines from an external point. The auxiliary circle intersects the original circle at two points, giving two tangent points.
✕ Point Inside Circle
Attempting the construction when P is inside or on the circle. A tangent from an external point requires \(OP > r\). If \(OP \le r\), no tangent from P exists.

Summary: Key Facts

Property Detail
Tangent \(\perp\) Radius \(\angle OTP = 90^\circ\) always
Number of tangents from external point Exactly 2
Equal tangent segments \(PT_1 = PT_2\)
Tangent length formula \(PT = \sqrt{OP^2 - r^2}\)
Construction tool Auxiliary circle with diameter \(\overline{OP}\)
Justification Thales' Theorem + Tangent-Radius Theorem

Question 1 of 5

A tangent to a circle is always perpendicular to which of the following?

Question 2 of 5

A circle has radius 9 and center O. Point P is 15 units from O. What is the length of the tangent segment from P to the circle?

Question 3 of 5

In the tangent construction, why do we draw a circle with diameter \(\overline{OP}\) rather than just any circle through O and P?

Question 4 of 5

Two tangent segments are drawn from external point P to a circle with center O and radius 8. If \(\angle T_1PT_2 = 60^\circ\), find \(OP\).

Question 5 of 5

Point P is at distance 4 from the center of a circle with radius 6. How many tangent lines from P to the circle can be constructed?