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Constructing
Tangent Lines to Circles

Learn to construct a tangent line from an external point to a circle using the properties of perpendicularity and semicircles.

Precalculus

What Is a Tangent Line?

\(O\)
\(T\)
radius
tangent line
\(l\)

A tangent line to a circle is a line that touches the circle at exactly one point, called the point of tangency. The tangent is always perpendicular to the radius drawn to the point of tangency.

Tangent from an External Point

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

Given a circle with center \(O\) and an external point \(P\) (outside the circle), there are always exactly two tangent lines from \(P\) to the circle. The segments \( PT_1 \) and \( PT_2 \) are called tangent segments.

Why Tangent Constructions Matter

Geometry Proofs
Tangent-radius perpendicularity is used in many circle theorem proofs and competition problems.
Engineering
Tangent lines model contact points in gears, wheels, and mechanical linkages.
Calculus Preview
The derivative at a point on a curve gives the slope of the tangent line: the core idea of differential calculus.

The Tangent-Radius Theorem

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

This "if and only if" relationship is the foundation for constructing tangent lines. It tells us that finding a tangent point is equivalent to finding a point where the radius meets a line at \( 90^\circ \).

Thales' Theorem: The Key Insight

auxiliary circle on \(OP\)
\(O\)
\(M\)
\(P\)
\(T\)

Thales' theorem states that any angle inscribed in a semicircle is a right angle. By constructing a circle with diameter \(OP\), every point on that circle "sees" the segment \( OP \) at \( 90^\circ \). The intersection with the original circle gives us the tangent point \(T\), where \( \angle OTP = 90^\circ \).

Tangent vs Secant vs External Line

Tangent
1 point
Secant
2 points
External
0 points
Tangent
Intersects the circle at exactly 1 point. Distance from center equals \( r \).
Secant
Intersects at exactly 2 points. Distance from center is less than \( r \).
External
Does not intersect. Distance from center is greater than \( r \).

Step-by-Step Construction

1
Draw segment \( OP \) from the center of the circle \( O \) to the external point \( P \).
2
Find the midpoint \( M \) of \( OP \) using a perpendicular bisector construction.
3
Draw a circle centered at \( M \) with radius \( MP \) (equivalently, \( MO \)). This is the auxiliary circle with diameter \( OP \).
4
Mark the intersections \( T_1 \) and \( T_2 \) of the auxiliary circle with the original circle.
5
Draw lines \( PT_1 \) and \( PT_2 \). These are the two tangent lines from \( P \) to the circle.

Why the Construction Works

1
\( T \) lies on the auxiliary circle with diameter \( OP \), so by Thales' theorem, \( \angle OTP = 90^\circ \).
2
\( T \) also lies on the original circle, so \( OT \) is a radius of the original circle.
3
Since \( OT \perp PT \) and \( OT \) is a radius, line \( PT \) is tangent to the circle at \( T \) by the tangent-radius theorem.

Example 1: Finding Tangent Length

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

Given: circle with center \(O\), radius \(r = 5\), and external point \(P\) with \(OP = 13\). Find the tangent length \( PT \).

Example 2: Coordinate Geometry

\(x\)
\(y\)
(3,4)
(8,4)
\(T_1\)
\(T_2\)
3
8
4

Circle centered at \((3, 4)\) with \(r = 2\). External point \(P = (8, 4)\). The tangent length is:

Example 3: Finding the Tangent Equation

Find the equation of the tangent lines from \( P(5, 0) \) to the circle \( x^2 + y^2 = 9 \).

1
Identify: center \( O = (0,0) \), radius \( r = 3 \), and \( OP = 5 \). Since \(5 > 3\), point \( P \) is external.
2
Tangent length: \( PT = \sqrt{25 - 9} = 4 \).
3
Line through \( P \): \( y = m(x - 5) \). For tangency, distance from \( O \) to this line equals \( r = 3 \).
4
Apply distance formula: \( \frac{|{-5m}|}{\sqrt{1 + m^2}} = 3 \implies 25m^2 = 9(1 + m^2) \implies m^2 = \frac{9}{16} \implies m = \pm \frac{3}{4} \).

Equal Tangent Segments Proof

\(O\)
\(P\)
\(T_1\)
\(T_2\)
\(a\)
\(a\)
≅
1
\( OT_1 = OT_2 = r \) (both radii of the same circle).
2
\( OP = OP \) (common side, shared hypotenuse).
3
\( \angle OT_1P = \angle OT_2P = 90^\circ \) (tangent-radius theorem).
4
By HL (Hypotenuse-Leg): \( \triangle OT_1P \cong \triangle OT_2P \), so \( PT_1 = PT_2 \). ✓

Connection to Power of a Point

The power of a point \( P \) with respect to a circle of center \( O \) and radius \( r \) is defined as:

\( \text{pow}(P) > 0 \)
\( P \) is outside the circle. Tangent length = \( \sqrt{\text{pow}(P)} \).
\( \text{pow}(P) = 0 \)
\( P \) is on the circle. The tangent is through \( P \) itself.
\( \text{pow}(P) < 0 \)
\( P \) is inside the circle. No tangent from \( P \) exists.

Common Mistakes to Avoid

✕ Wrong Right Angle
Assuming the right angle is at P instead of at the tangent point T. The \( 90^\circ \) angle is always where the radius meets the tangent.
✕ Adding Instead of Subtracting
Using \( PT = \sqrt{OP^2 + r^2} \) instead of \( \sqrt{OP^2 - r^2} \). The radius is a leg, not added to the hypotenuse.
✕ Internal Point
Attempting to draw a tangent from a point inside the circle. Always verify that \(OP > r\) before constructing.

Summary: Tangent Line Properties

Property Formula / Result
Perpendicularity \( OT \perp l \) Tangent-radius theorem
Tangent length \( PT = \sqrt{OP^2 - r^2} \) Pythagorean theorem
Equal segments \( PT_1 = PT_2 \) HL congruence
Construction tool Auxiliary circle on \( OP \) Thales' theorem
External point test \(OP > r\) \(\text{pow}(P) > 0\)

Question 1 of 5

A circle has center \( O \) and radius \( 6 \). Point \( P \) is outside the circle with \( OP = 10 \). What is the length of a tangent segment from \( P \) to the circle?

Question 2 of 5

In the tangent line construction, the auxiliary circle is drawn with what as its diameter?

Question 3 of 5

Why does the construction guarantee that \( \angle OTP = 90^\circ \)?

Question 4 of 5

A circle has equation \( x^2 + y^2 = 25 \). From the point \( P(7, 1) \), the tangent length to the circle is:

Question 5 of 5

If \( PT_1 \) and \( PT_2 \) are both tangent segments from the same external point \( P \) to a circle, which of the following is always true?