Exercise 9.1 — Length of Tangent

Length of the tangent drawn from an external point to a circle.

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Exercise 9.1 — Tangents and Secants to a Circle

Exercise 9.1 opens Chapter 9, Tangents and Secants to a Circle, in Class 10 Mathematics for students following the CBSE, Telangana, and Andhra Pradesh board syllabi. This exercise blends quick conceptual recall (fill-in-the-blanks), direct numerical applications of the tangent-length formula, a construction-based question, and a complete geometric proof about parallel tangents.

It's the perfect bridge between the theory introduced at the start of this chapter and the more advanced proof-based exercises that follow, since it tests whether you've internalized the basic vocabulary and the tangent-radius perpendicularity property before applying them to real problems.

Secant vs Tangent Length of Tangent Formula Tangent-Radius Perpendicularity Parallel Tangents at Diameter Ends

CORE FORMULA

If O is the center of a circle with radius r, and P is an external point at distance d from O, then the length of the tangent from P to the circle is: PQ = √(d² − r²)

Question 1 — Fill in the Blanks (Key Concepts Recap)

This question checks your grasp of the fundamental vocabulary introduced for tangents and secants. Each blank tests a specific definition or property that forms the basis of every later proof in this chapter.

Part	Statement	Correct Answer
(i)	A tangent to a circle intersects it in _____ points.	One
(ii)	A line intersecting a circle in two points is called a _____.	Secant
(iii)	A circle can have _____ parallel tangents at the most.	Two
(iv)	The common point of a tangent to a circle and the circle is called the _____.	Point of contact
(v)	We can draw _____ tangents to a circle.	Infinite

💡 Why "infinite" tangents but only "two" parallel ones: A circle has a tangent line at every single point on its boundary, which means there are infinitely many tangents in total. However, for any chosen direction, only two tangent lines can be drawn parallel to each other — one touching the circle on each opposite side, like two parallel rulers touching a circular plate from top and bottom.

Question 2 — Find the Length of Tangent PQ (Radius 5 cm, OQ = 13 cm)

A tangent PQ at point P on a circle of radius 5 cm meets a line through the center O at point Q, where OQ = 13 cm. We need to find the length of PQ — a direct application of the tangent length formula.

Right Triangle OPQ

∠OPQ = 90°, OQ = 13 cm (hypotenuse)

Question 2 — Solution

Find PQ, given radius OP = 5 cm and OQ = 13 cm

Given: OP (radius) = 5 cm, OQ = 13 cm Since PQ is tangent at P, OP ⊥ PQ ⟹ Triangle OPQ is right-angled at P Apply Pythagoras Theorem: OQ² = OP² + PQ² 13² = 5² + PQ² 169 = 25 + PQ² PQ² = 169 − 25 = 144 PQ = √144 ∴ PQ = 12 cm

Radius (OP)

5 cm

Given

13 cm

Given

12 cm

Tangent length

📌 Spot the Pythagorean triple: The numbers 5, 12, 13 form one of the most common Pythagorean triples used in board exam questions. Recognizing this pattern (5² + 12² = 13²) can help you verify your answer quickly without recalculating the square root from scratch.

Question 3 — Draw a Circle with a Parallel Tangent and a Parallel Secant

This is a construction-based question: draw a circle, and then draw two lines parallel to a given outside line, such that one of them is a tangent to the circle and the other is a secant.

Question 3 — Construction Steps

Construct parallel tangent and secant lines to a given line

Step 1: Draw a circle with any convenient center O and radius, and draw the given line outside the circle.
Step 2: To draw the parallel tangent, draw a line parallel to the given line that just touches the circle at exactly one point — this line should be positioned exactly at a perpendicular distance equal to the radius from the center.
Step 3: To draw the parallel secant, draw another line parallel to the given line, but positioned closer to the center so that it cuts through the circle's interior, crossing the boundary at two distinct points.
Step 4: Label the point of contact on the tangent line, and label the two intersection points on the secant line.

✅ Key construction insight: Whether a parallel line becomes a tangent or a secant depends entirely on its perpendicular distance from the center. If that distance equals the radius, the line is tangent. If the distance is less than the radius, the line is a secant. If the distance is greater than the radius, the line doesn't touch the circle at all.

Question 4 — Calculate Tangent Length (External Point 15 cm, Radius 9 cm)

Find the length of the tangent drawn from a point 15 cm away from the center of a circle of radius 9 cm. This is the same tangent-length formula applied with a different Pythagorean triple.

Question 4 — Solution

Find tangent length PQ, given OP = 15 cm and radius OQ = 9 cm

Given: O = center, P = external point with OP = 15 cm Tangent from P touches the circle at Q, so OQ (radius) = 9 cm Since OQ ⊥ PQ, triangle OQP is right-angled at Q Apply Pythagoras Theorem: OP² = OQ² + PQ² 15² = 9² + PQ² 225 = 81 + PQ² PQ² = 225 − 81 = 144 PQ = √144 ∴ PQ = 12 cm

Distance (OP)

15 cm

Given

Radius (OQ)

9 cm

Given

Tangent (PQ)

12 cm

Calculated

💡 Another Pythagorean triple: Notice that 9, 12, 15 is simply the 3-4-5 triple scaled by 3 (since 3×3=9, 4×3=12, 5×3=15). Recognizing scaled versions of common triples can save valuable time during board exams.

Question 5 — Prove That Tangents at the Ends of a Diameter Are Parallel

This question asks for a formal proof that the two tangents drawn at the two endpoints of a diameter are always parallel to each other. The proof relies entirely on the tangent-radius perpendicularity theorem combined with the co-interior angle test for parallel lines.

Tangents PQ and RS at A and B

AB is the diameter; PQ ∥ RS

Question 5 — Proof

Prove tangents PQ (at A) and RS (at B) are parallel, where AB is a diameter

Given: O is the center, AB is a diameter PQ is the tangent at A; RS is the tangent at B Step 1 — Apply the tangent-radius theorem at both points: ∠OAQ = 90° (tangent ⊥ radius at A) ∠OBS = 90° (tangent ⊥ radius at B) Step 2 — Add the two angles: ∠OAQ + ∠OBS = 90° + 90° = 180° Step 3 — Apply the co-interior angle test: Since AB is a transversal cutting both tangent lines, and the sum of the co-interior (consecutive interior) angles on the same side is 180°, PQ ∥ RS (by the converse of the co-interior angles theorem) ∴ The tangents to a circle at the endpoints of a diameter are parallel ✓

✅ Why this works so cleanly: Both tangent lines are perpendicular to the same straight line — the diameter AB acts as a common transversal that is perpendicular to both tangents. Two lines that are both perpendicular to the same line are always parallel to each other; this is really just a special, elegant case of that broader geometric fact.

Key Theorems and Formulas Used in This Exercise

Theorem / Formula	Statement	Used In
Tangent ⊥ Radius	The tangent at any point is perpendicular to the radius through that point	Q2, Q4, Q5
Length of Tangent Formula	PQ = √(OP² − OQ²), derived from Pythagoras Theorem	Q2, Q4
Perpendicular Distance Rule	A line parallel to a given line is tangent if its distance from the center equals the radius; secant if less	Q3
Co-interior Angles Theorem	If co-interior angles formed by a transversal sum to 180°, the two lines are parallel	Q5

Common Mistakes to Avoid

Mixing up which side is the hypotenuse: In tangent length problems, the line from the external point to the center (OP or OQ, whichever connects to the center) is always the hypotenuse — never the radius or the tangent itself. Misidentifying the hypotenuse leads to an incorrect equation.
Saying "infinite parallel tangents" instead of "two": While a circle has infinitely many tangents overall, only two of them can ever be parallel to each other for any given direction. This is a frequently confused point in fill-in-the-blank questions.
Forgetting to justify the right angle in proofs: In Question 5, simply stating ∠OAQ = 90° without explicitly citing the "tangent is perpendicular to radius" theorem can cost marks in board exams — always state the reason.
Confusing secant and tangent during construction: In Question 3, remember a secant must clearly cross the circle's boundary at two visibly separate points, while the tangent line should only touch the circle at a single point without entering its interior.

⛔ Exam trap: When using the co-interior angle test in Question 5, students sometimes forget to explicitly mention that AB is acting as the transversal connecting the two tangent lines. Always name the transversal clearly before stating the parallel lines conclusion.

Quick Reference — All Answers at a Glance

Question	What Was Asked	Answer
Q1(i)	Tangent intersects circle in how many points?	One
Q1(ii)	Line intersecting circle in two points is called	Secant
Q1(iii)	Maximum number of parallel tangents	Two
Q1(iv)	Common point of tangent and circle	Point of contact
Q1(v)	Total number of tangents possible to a circle	Infinite
Q2	PQ, given radius=5cm, OQ=13cm	12 cm
Q3	Construct parallel tangent & secant	Construction (see steps above)
Q4	Tangent length, OP=15cm, radius=9cm	12 cm
Q5	Prove tangents at diameter ends are parallel	Proved using 90°+90°=180°

What This Exercise Prepares You For

Exercise 9.1 builds the essential toolkit — vocabulary, the tangent length formula, and the tangent-radius perpendicularity property — that every later exercise in Chapter 9 depends on, including problems involving tangents drawn from an external point and the equal tangent segments theorem. This is a heavily tested section in Telangana SSC, AP SSC, and CBSE Class 10 board exams, especially the fill-in-the-blank style questions and the 12 cm tangent-length pattern shown in Q2 and Q4.

To reinforce the prerequisite skills used here, review the Introduction to Tangents and Secants for the tangent-radius theorem derivation, and Exercise 8.4 — Pythagoras Theorem, since every tangent-length calculation in this exercise is a direct Pythagoras Theorem application.

📐 Board Exam Strategy (Telangana & AP SSC, CBSE): Fill-in-the-blank questions like Q1 are common 1-mark questions, while tangent length problems like Q2 and Q4 frequently appear as 2-mark questions. For full marks on proof questions like Q5, always explicitly state the tangent-radius perpendicularity theorem as your reason before applying any angle-sum or parallel-line argument.