How to Solve: Tangent Problems

How to Solve: Tangent Problems

(Geometry – For Students & Teachers)

Introduction

"Mastering tangents doesn’t just help you ace geometry—it’s how engineers design roller coasters, architects build bridges, and GPS calculates the fastest route. Today, you’ll learn the exact steps to solve any tangent problem in under 60 seconds."

What You Need To Know First

Before tackling tangents, ensure you understand: 1. Circle Properties – Radius, diameter, chord, secant, and the fact that a radius is perpendicular to a tangent at the point of tangency. 2. Right Triangles & Pythagoras’ Theorem – You’ll use this to find missing lengths. 3. Similar Triangles – Some tangent problems involve overlapping triangles with proportional sides.

Key Vocabulary

Formulas To Know

Step-by-Step Method

Follow these steps for every tangent problem:

1. Draw the Diagram – Sketch the circle, tangent(s), and any given points. Label everything.
2. Identify the Right Angle – Mark the 90° angle where the radius meets the tangent.
3. Label Known Lengths – Write down all given measurements (radii, tangents, secants).
4. Apply the Correct Theorem –
5. If two tangents from one point: Two-Tangent Theorem ((PA = PB)).
6. If one tangent and one secant: Tangent-Secant Theorem ((PA^2 = PB \times PC)).
7. If only a tangent and radius: Pythagoras’ Theorem (since radius ⊥ tangent).
8. Set Up the Equation – Plug known values into the formula.
9. Solve for the Unknown – Use algebra (cross-multiply, square roots, etc.).
10. Check Units & Reasonableness – Does the answer make sense? (e.g., a tangent can’t be negative).

Worked Example Using the Steps

Problem: A tangent (PT) touches a circle at (T). The radius (OT) is 5 cm, and the distance from the external point (P) to the center (O) is 13 cm. Find the length of (PT).

Solution: 1. Draw the Diagram – Circle with center (O), tangent (PT), radius (OT), and line (OP). 2. Identify the Right Angle – (OT \perp PT) (radius ⊥ tangent), so (\angle OTP = 90°). 3. Label Known Lengths – (OT = 5) cm, (OP = 13) cm. 4. Apply the Correct Theorem – Right triangle (OTP), so use Pythagoras’ Theorem. 5. Set Up the Equation – (OT^2 + PT^2 = OP^2) → (5^2 + PT^2 = 13^2). 6. Solve for the Unknown –
(25 + PT^2 = 169)
(PT^2 = 169 - 25 = 144)
(PT = \sqrt{144} = 12) cm. 7. Check – 12 cm is reasonable (less than 13 cm, as expected).

Answer: (PT = 12) cm.

Worked Examples

Example 1 – Basic (Two-Tangent Theorem)

Problem: Two tangents (PA) and (PB) are drawn from an external point (P) to a circle with center (O). If (PA = 8) cm, find (PB).

Solution: 1. Draw the circle, tangents (PA) and (PB), and label (P), (A), (B). 2. Recall the Two-Tangent Theorem: (PA = PB). 3. Since (PA = 8) cm, (PB = 8) cm.

What we did and why: The Two-Tangent Theorem guarantees that tangents from the same external point are equal. No calculations needed—just apply the rule.

Answer: (PB = 8) cm.

Example 2 – Medium (Tangent-Secant Theorem)

Problem: From point (P) outside a circle, a tangent (PA) and a secant (PBC) are drawn. If (PA = 6) cm and (PB = 4) cm, find (BC).

Solution: 1. Draw the circle, tangent (PA), and secant (PBC) (with (B) between (P) and (C)). 2. Label (PA = 6) cm, (PB = 4) cm. 3. Apply the Tangent-Secant Theorem: (PA^2 = PB \times PC). 4. (PC = PB + BC), so let (BC = x). Then (PC = 4 + x). 5. Set up the equation: (6^2 = 4 \times (4 + x)). 6. Solve:
(36 = 16 + 4x)
(20 = 4x)
(x = 5) cm.

What we did and why: The Tangent-Secant Theorem relates the tangent to the secant’s parts. We substituted (PC) in terms of (x) and solved for (BC).

Answer: (BC = 5) cm.

Example 3 – Exam Style (Disguised Problem)

Problem: A circular garden has a radius of 7 m. A straight path is tangent to the garden at point (T). If the path is 24 m from the center (O) of the garden, how long is the path between the point of tangency (T) and the point (P) where the path meets a fence 25 m from (O)?

Solution: 1. Understand the Problem – The path is tangent at (T), and (OP = 25) m. We need (PT). 2. Draw the Diagram – Circle with center (O), tangent (PT), radius (OT = 7) m, and (OP = 25) m. 3. Identify the Right Angle – (OT \perp PT) (radius ⊥ tangent). 4. Use Pythagoras’ Theorem – In right triangle (OTP):
(OT^2 + PT^2 = OP^2). 5. Plug in values:
(7^2 + PT^2 = 25^2)
(49 + PT^2 = 625)
(PT^2 = 576)
(PT = 24) m.

What we did and why: The problem disguises the tangent-radius right triangle. Recognizing the 90° angle lets us use Pythagoras’ Theorem directly.

Answer: (PT = 24) m.

Common Mistakes

Exam Traps

1-Minute Recap

"Alright, listen up—this is your 60-second tangent survival guide. First, draw the diagram—always. Second, mark the right angle where the radius hits the tangent. Third, pick the right theorem: - Two tangents from one point? They’re equal—Two-Tangent Theorem. - One tangent and one secant? Square the tangent, set it equal to the secant’s parts—Tangent-Secant Theorem. - Just a tangent and radius? Pythagoras’ Theorem—it’s a right triangle. Write the equation, solve, and check your answer. If it’s negative or longer than the hypotenuse, you messed up. Now go crush that exam!