Tangent Lines

📏 Tangent Lines

Part 1 of 5 — The Tangent and Its Right Angle

Topics in This Part

🔑 Key Concept: A tangent line touches a circle at exactly one point. The single most useful fact about it: a tangent is always perpendicular to the radius drawn to that point of contact.

Tangent vs. Secant

A line and a circle can meet in three ways:

The point where a tangent touches the circle is the point of tangency (or point of contact).

💡 The word tangent comes from the Latin tangere, "to touch." A tangent line touches but never crosses into the circle.

Concept Check 🎯

The Radius–Tangent Theorem

This is the rule the entire topic is built on:

🔑 Tangent–Radius Theorem: A tangent line is perpendicular to the radius drawn to the point of tangency.

If circle $O$ has a tangent line touching at point $T$, and $P$ is any other point on that tangent line, then radius $\overline{OT}$ meets the tangent at a right angle:

$\overline{OT} \perp \text{tangent line} \quad\Longrightarrow\quad \angle OTP = 90°$

Why it matters

That right angle lets you build a right triangle any time a tangent appears — which means the Pythagorean theorem is suddenly available. Watch:

If $P$ is a point outside the circle, $T$ is a point of tangency, and $O$ is the center, then $\triangle PTO$ has a right angle at $T$:

$PT^2 + r^2 = PO^2$

where $r = OT$ is the radius.

⚠️ The right angle is at the point of tangency $T$ — not at the center and not at the external point. So the hypotenuse is always $PO$ (center to outside point).

Use the Right Angle 🧮

In each diagram, $\overline{PT}$ is tangent to circle $O$ at $T$, with radius $r = OT$ and $PO$ the distance from the center to $P$. Use $PT^2 + r^2 = PO^2$.

1) $r = 5$, $PO = 13$. Find $PT$. 2) $r = 6$, $PO = 10$. Find $PT$. 3) $r = 8$, $PO = 17$. Find $PT$.

Naming What You See

Before moving on, make sure you can name a line by how it meets a circle:

• $0$ shared points → the line misses the circle entirely.
• $1$ shared point → tangent.
• $2$ shared points → secant.

A chord is the segment joining the two crossing points of a secant, and the diameter is the longest chord (it passes through the center).

Classify the Line 🔽

Match each description to the correct name.

Wrap-Up

You now know the two facts the whole topic leans on:

1. A tangent touches a circle at exactly one point.
2. The radius to that point is perpendicular to the tangent — giving you a right triangle and the Pythagorean theorem.

💡 Whenever you see a tangent in a problem, your first move should be: draw the radius to the point of tangency and mark the right angle.

In Part 2 we use this to compare two tangents drawn from the same outside point.

📏 Tangent Lines

Part 2 of 5 — Tangent Segments from an External Point

🔑 The Big Idea: From a single point outside a circle you can draw exactly two tangent lines, and the two tangent segments are congruent — they have equal length.

Two Tangents, Equal Lengths

From an external point $P$, draw the two tangent segments $\overline{PA}$ and $\overline{PB}$, touching the circle at $A$ and $B$.

🔑 Two-Tangent Theorem: Tangent segments drawn to a circle from the same external point are congruent: $PA = PB$.

Why it's true

Connect the center $O$ to $P$, $A$, and $B$. You get two right triangles, $\triangle PAO$ and $\triangle PBO$:

• $\angle PAO = \angle PBO = 90°$ (radius ⊥ tangent)
• $OA = OB = r$ (both are radii)
• $PO = PO$ (shared hypotenuse)

So $\triangle PAO \cong \triangle PBO$ (Hypotenuse–Leg), which forces $PA = PB$. ✓

Concept Check 🎯

A Classic Application: Inscribed Polygons

When a circle is inscribed in a triangle (or any polygon), each side is tangent to the circle. The two-tangent theorem says the tangent segments from each vertex are equal — that lets you find unknown lengths.

Worked Example

A circle is inscribed in a triangle. The tangent lengths from the three vertices are $x$, $y$, and $z$, so each side equals the sum of two tangent lengths. If the tangent segments from one vertex measure $5$ and from the next vertex measure $7$, then the side between them is:

$5 + 7 = 12$

💡 At every vertex, the two tangent segments leaving that vertex are equal. Label them once and reuse the labels around the whole figure.

Tangent-Length Problems 🧮

1) $\overline{PA}$ and $\overline{PB}$ are tangent from $P$. If $PA = 3x + 4$ and $PB = 5x - 8$, the segments are equal. Solve for $x$. 2) Using the value of $x$ from #1, find the common tangent length $PA$. 3) Tangents from $P$ have length $10$. The radius is $r$ and $PO = 26$. Find $r$.

Setting Up an Inscribed-Circle Side

Picture a triangle with an inscribed circle. Call the tangent lengths from the three vertices $a$, $b$, and $c$. Each side is the sum of the two tangent lengths from its endpoints:

$\text{side} = (\text{tangent from one vertex}) + (\text{tangent from the other vertex})$

So if $a = 4$ and $b = 9$, the side joining those two vertices is $4 + 9 = 13$.

Inscribed-Circle Sides 🔽

A triangle has an inscribed circle with tangent lengths $a = 5$, $b = 8$, and $c = 3$ from its three vertices.

Wrap-Up

Two tangents from one external point are always equal in length — proven by Hypotenuse–Leg. This single fact unlocks a huge family of problems: inscribed circles, kite-shaped figures, and "find $x$" tangent equations.

⚠️ The theorem applies only when both tangents come from the same external point. Tangents from two different points are generally not equal.

Next, in Part 3, we measure the angles that tangents make with chords and with each other.

📏 Tangent Lines

Part 3 of 5 — Tangent Angles & Intercepted Arcs

🔑 The Theme: Angles formed by tangents are measured by the arcs they intercept. Two formulas do all the work: the tangent–chord angle and the two-tangent angle.

The Tangent–Chord Angle

When a tangent and a chord meet at the point of tangency, the angle between them is half the intercepted arc.

🔑 Tangent–Chord Theorem: $\text{angle} = \dfrac{1}{2}(\text{intercepted arc})$

Worked Example

A chord meets a tangent at the point of tangency, intercepting an arc of $140°$. The tangent–chord angle is:

$\frac{1}{2}(140°) = 70°$

The chord cuts the circle into two arcs that add to $360°$, so the other arc is $360° - 140° = 220°$, and the angle on the other side of the chord is:

$\frac{1}{2}(220°) = 110°$

💡 Check: $70° + 110° = 180°$, exactly a straight line along the tangent. ✓

Tangent–Chord Practice 🧮

A chord meets a tangent at the point of tangency. Find each tangent–chord angle (in degrees — enter the number only).

1) Intercepted arc $= 80°$. Angle $= \,?$ 2) Intercepted arc $= 200°$. Angle $= \,?$ 3) The tangent–chord angle is $35°$. Find its intercepted arc.

The Two-Tangent Angle

When two tangents meet at an external point $P$, they intercept a near arc (closer to $P$) and a far arc (across the circle). The angle at $P$ is half the difference:

$\angle P = \frac{1}{2}\big(\text{far arc} - \text{near arc}\big)$

Because the two arcs make a full circle, $\text{far arc} + \text{near arc} = 360°$.

Worked Example

Two tangents from $P$ intercept a near arc of $100°$. Then the far arc is $360° - 100° = 260°$, and:

$\angle P = \frac{1}{2}(260° - 100°) = \frac{1}{2}(160°) = 80°$

💡 Shortcut: The angle at $P$ and the near arc are supplementary: $\angle P + (\text{near arc}) = 180°$. Here $80° + 100° = 180°$. ✓ (This comes from the quadrilateral $PAOB$ having two $90°$ angles.)

Two-Tangent Angle Walkthrough 🔽

Two tangents from external point $P$ intercept a near arc of $120°$. Work through the angle at $P$.

Two Formulas, One Pattern

Both tangent-angle rules say the same thing: an angle is half of an arc (or half of an arc difference).

💡 Vertex on the circle → use one arc. Vertex outside → use the difference of the two arcs.

Concept Check 🎯

📏 Tangent Lines

Part 4 of 5 — Tangents on the Coordinate Plane

🔑 Goal: Put tangents on the $xy$-plane. We'll find the slope of a tangent line and test whether a given line is tangent to a circle.

The Tangent Is Perpendicular to the Radius

On a coordinate grid, the radius–tangent right angle becomes a slope statement:

🔑 The tangent's slope is the negative reciprocal of the radius's slope (because perpendicular slopes multiply to $-1$).

Worked Example

Circle centered at $O = (0, 0)$, with point of tangency $T = (3, 4)$ on the circle.

Step 1 — slope of the radius $\overline{OT}$: $m_{\text{radius}} = \frac{4 - 0}{3 - 0} = \frac{4}{3}$

Step 2 — slope of the tangent (negative reciprocal): $m_{\text{tangent}} = -\frac{3}{4}$

Step 3 — equation through $T = (3,4)$ with slope $-\frac{3}{4}$: $y - 4 = -\frac{3}{4}(x - 3)$

💡 Check perpendicularity: $\dfrac{4}{3} \cdot \left(-\dfrac{3}{4}\right) = -1$. ✓ Perpendicular slopes always multiply to $-1$.

Slopes of Tangents 🧮

A circle is centered at the origin $O=(0,0)$. For each point of tangency $T$, find the slope of the tangent line (negative reciprocal of the radius slope). Enter as a fraction like -2/5 or a whole number.

1) $T = (2, 6)$. Radius slope $= 3$, so tangent slope $= \,?$ 2) $T = (5, -2)$. Tangent slope $= \,?$ 3) $T = (0, 7)$ (top of the circle). Tangent slope $= \,?$

Is a Line Tangent? The Distance Test

A line is tangent to a circle exactly when its distance from the center equals the radius:

Worked Example

Is the segment with $PT = 12$, radius $r = 5$, and external distance $PO = 13$ a valid tangent setup?

$PT^2 + r^2 = 12^2 + 5^2 = 144 + 25 = 169 = 13^2 = PO^2 \;✓$

Because the Pythagorean relationship holds with the right angle at $T$, the line through $P$ and $T$ is genuinely tangent.

⚠️ If $PT^2 + r^2 \ne PO^2$, the line is not tangent — the angle at $T$ wouldn't be $90°$.

Tangent or Not? 🔽

A circle has radius $r$. For each line, compare its distance $d$ from the center to $r$.

Tying the Two Tools Together

On the coordinate plane, tangent problems reduce to two ideas:

1. Slope: the tangent's slope is the negative reciprocal of the radius's slope (they meet at $90°$).
2. Distance: a line is tangent exactly when its distance from the center equals $r$.

💡 Both are just the perpendicular-radius fact in disguise — one as a slope statement, one as a distance statement.

Concept Check 🎯

📏 Tangent Lines

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) use the radius–tangent right angle, (2) apply the two-tangent theorem, (3) measure tangent angles by their arcs, and (4) work tangents on the coordinate plane. Let's put it together.

Quick Reference

⚠️ Two reminders: the right angle is at the point of tangency, and the two-tangent theorem needs both tangents from the same external point.

Mixed Practice 🎯

A Solving Game Plan

When a tangent problem appears, run this checklist:

1. Mark the right angle at every point of tangency and draw the radius there.
2. Look for two tangents from one point → set their lengths equal.
3. Need an angle? Find the intercepted arc, then take half of it (or half the difference).
4. On a grid? Use perpendicular slopes (negative reciprocals) or the distance-equals-radius test.

💡 Almost every tangent problem is one of these four moves — or a short combination of them.

Mixed Numeric Practice 🧮

1) Tangents $\overline{PA}$ and $\overline{PB}$ from $P$: $PA = 4x - 3$ and $PB = x + 9$. Solve for $x$. 2) A tangent–chord angle measures $63°$. Find its intercepted arc (degrees). 3) $\overline{PT}$ tangent at $T$, $PO = 20$, $PT = 16$. Find the radius $r$.

One Last Look

You've used every tangent tool in this lesson. The exit quiz below pulls from all five parts: the right angle, the two-tangent theorem, and the distance test.

🔑 Remember the anchor fact: a tangent is perpendicular to the radius at the point of contact. Almost everything else follows from it.

Exit Quiz ✅

Answer all three to finish the lesson.