1. Getting Friendly With the Circle 🟢

Before you crack any Circles Quiz, you must be comfortable with the basic language of a circle. These terms seem simple, but many silly mistakes in exams come from confusing them.

Core terms at a glance

When you read questions in your Circles Quiz, underline words like radius, chord, tangent, point of contact, secant. They tell you which theorem is likely to appear.

2. Must-Know Theorems for Class 10 Circles 📚

These theorems form the backbone of almost every board exam and school test question.

Theorem 1: Radius is perpendicular to tangent

The tangent at any point of a circle is perpendicular to the radius through the point of contact.

If $O$ is the centre, $P$ is the point of contact, and $PQ$ is the tangent, then:

Use it when you see: “PQ is a tangent to the circle at P” and “O is the centre”. Immediately think of a right angle at P.

Theorem 2: Tangents from an external point are equal

The lengths of tangents drawn from an external point to a circle are equal.

If $PA$ and $PB$ are tangents from an external point $P$ to a circle with centre $O$:

This simple fact helps you find missing lengths and prove triangles congruent.

Theorem 3: Equal chords, equal distances from centre

Equal chords of a circle are equidistant from the centre and vice versa.

If chord $AB$ and chord $CD$ are equal, then perpendicular distances from centre $O$ to these chords are also equal.

Theorem 4: Angle subtended by chord at centre and at circumference

The angle subtended by a chord at the centre is twice the angle subtended at a point on the remaining part of the circle.

If chord $AB$ subtends angle $\angle AOB$ at the centre and $\angle ACB$ at a point $C$ on the circle, then:

Theorem 5: Cyclic quadrilateral property

The sum of opposite angles of a cyclic quadrilateral is $180^\circ$.

If $ABCD$ is a quadrilateral with all its vertices on a circle, then:

and

3. Step-by-Step Example Practice 🧮

Let’s work through a typical Circles Quiz problem, exactly the style you see in Class 10 board exams.

Example 1: Using tangent-radius perpendicularity

Question:
A circle has centre O. A tangent at point P to the circle meets a line through O at point Q, such that $OP = 6$ cm and $OQ = 10$ cm. Find the length of $PQ$.

Step 1: Visualise and mark right angle

Since $PQ$ is a tangent at $P$ and $OP$ is the radius, we know:

So triangle $OPQ$ is right-angled at $P$.

Step 2: Recognise given sides

You know:

• $OP = 6$ cm (radius)
• $OQ = 10$ cm (hypotenuse)

We need $PQ$ (the other side of the right triangle).

Step 3: Apply Pythagoras theorem

In right triangle $OPQ$:

Substitute values:

Final answer:
The length of the tangent $PQ$ is 8 cm.

Example 2: Tangents from an external point

Question:
From a point P outside a circle with centre O, two tangents PA and PB are drawn to the circle. If $PA = 12$ cm and $OP = 13$ cm, find the radius of the circle.

Step 1: Understand the right triangle

Radius $OA$ is perpendicular to tangent $PA$ at $A$:

So in triangle $OAP$:

• Hypotenuse = $OP = 13$ cm
• One side (tangent) = $PA = 12$ cm
• Other side = radius $OA = r$ (unknown)

Step 2: Apply Pythagoras theorem

Substitute:

Final answer:
The radius of the circle is 5 cm.

4. Fast Revision Snapshot for Exams ⚡

Use this mini-checklist just before a Circles Quiz or CBSE Class 10 Board exam:

• Tangent and radius at point of contact are perpendicular.
• Tangents from an external point are equal in length.
• Equal chords subtend equal angles at the centre.
• Angle at centre is double the angle at circumference (same chord).
• Opposite angles of a cyclic quadrilateral add up to 180°.
• Perpendicular from centre to chord bisects the chord.
• If two chords are equal, their distances from centre are equal.

Try to say these out loud once. It improves retention during the actual test.

5. Common Traps and Mistakes in Circles Questions 🚨

Students often know the theorems but still lose marks because of small errors.

Mistake 1: Forgetting the right angle at the point of contact

Many students don’t mark $\angle OPA = 90^\circ$ when PA is tangent.

Fix:
The moment you see the word “tangent” and “radius”, quickly draw the right angle symbol in your diagram.

Mistake 2: Confusing secant and tangent

• Tangent touches at one point.
• Secant cuts at two points.

Using tangent theorems on a secant will give wrong answers.

Mistake 3: Ignoring which angle is subtended at the centre

If a chord makes angle $\theta$ at the centre, the angle at the circumference on the same side is $\theta/2$, not the other way round.

Always remember:

Mistake 4: Not drawing auxiliary lines

In many CBSE and school test questions, the figure given is not enough. You may need to draw:

• A radius to the point of contact
• A perpendicular from centre to chord
• A diameter to use cyclic quadrilateral property

If you’re stuck, ask yourself: “Can I add a radius or perpendicular here?”

6. Strategy to Crack Circles Quiz Questions 🎯

Here is a simple 4-step method you can follow during any test or board exam.

Step 1: Decode the keywords

Underline words like:

• Tangent
• Radius
• Chord
• Cyclic quadrilateral
• Point of contact
• External point

Once you underline them, your brain starts matching them to the correct theorems.

Step 2: Mark all right angles and equal lengths

From theory:

• Wherever radius meets tangent → mark 90°
• From external point P, tangents PA and PB → mark PA = PB
• Perpendicular from centre to chord → mark equal halves of chord

These markings often give you pairs of congruent triangles.

Step 3: Choose the correct theorem or tool

Common tools:

• Pythagoras theorem (whenever you see a right angle and two sides)
• Tangent properties (equal tangents, perpendicular radius)
• Cyclic quadrilateral angle sum
• Angle at centre vs angle at circumference

Step 4: Present solution clearly

In board exams, even if your final answer is wrong, clear steps can still fetch partial marks. Always:

• Write which theorem you are using.
• Mention reason with each step (e.g., “since tangents from an external point are equal”).

7. Mixed Practice: Try These on Your Own 📝

Try to solve these without looking at the solution, as if you are in a real Circles Quiz.

1. From an external point P, two tangents PA and PB are drawn to a circle with centre O. If $PA = 7$ cm, find the perimeter of triangle PAB.

2. In a circle with centre O, chord AB subtends an angle of $120^\circ$ at the centre. Find the angle subtended by chord AB at a point C on the remaining part of the circle.

3. In a cyclic quadrilateral ABCD, if $\angle A = 95^\circ$, find $\angle C$.

4. A chord of a circle is 16 cm long and is at a distance of 15 cm from the centre. Find the radius of the circle.

For Q4, use the idea that the perpendicular from the centre to the chord bisects the chord, then apply Pythagoras.

8. Concept Map: Connecting All Ideas in Circles 🧠

Think of “Circles” as one big mind map:

• Tangent
  • Perpendicular to radius
  • Equal from an external point
• Chord
  • Perpendicular from centre → bisects chord
  • Equal chords ↔ equal distance from centre
• Angles
  • Angle at centre = 2 × angle at circumference
  • Same segment theorem (angles in same segment are equal)
• Cyclic Quadrilateral
  • Opposite angles supplementary
  • Exterior angle = interior opposite angle

When you see a problem, quickly identify which branch it belongs to. This mental mapping is very useful for JEE foundation and NTSE-level geometry as well.

9. Quick Board-Exam Style Question With Full Solution 🧾

Question:
In the given figure, PA and PB are tangents drawn from an external point P to a circle with centre O. If $\angle APB = 70^\circ$, find $\angle AOB$.

Step 1: Recognise equal tangents

From P, two tangents PA and PB are drawn to the circle. So:

Therefore, triangle APB is isosceles. Thus:

Let each of these angles be $x$.

Step 2: Use angle sum of triangle

In triangle APB:

So:

So:

Step 3: Use radius-tangent perpendicularity

OA and OB are radii, and both are perpendicular to the tangents at A and B respectively. So:

In triangle OAP, we know:

• $\angle OAP = 90^\circ$
• $\angle PAB = 55^\circ$

So:

Similarly, in triangle OBP:

Step 4: Find central angle AOB

Angle AOB is the full angle at centre between OA and OB:

Final answer:
$\angle AOB = 70^\circ$.

Notice how angle at the centre is equal to angle APB here because of the special configuration of tangents.

10. Final Revision Nuggets Before You Attempt the Circles Quiz 🧩

• Draw neat, labelled diagrams for every question.
• Highlight right angles and equal tangents as soon as you spot them.
• Always mention the theorem used: it fetches marks even if arithmetic slips.
• Practise a mix of:
  • Direct theorem questions
  • Prove-type questions
  • Numerical problems using Pythagoras

A strong grip on circles in Class 10 will make later geometry (especially coordinate geometry and trigonometry applications in higher classes and competitive exams) much easier.