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West Bengal Board (WBBSE) Class 10 (Madhyamik) Mathematics Set 01 (Foundation Level) English Medium 100% Free Practice Test

West Bengal Board (WBBSE) Class 10 (Madhyamik) Mathematics: Construction of Circumcircle and Incircle of a Triangle Online Test (Set 01 (Foundation Level))

Construction of Circumcircle and Incircle of a Triangle — Set 01 (Foundation Level)

Practise free online CBT mock test for West Bengal Board (WBBSE) Class 10 (Madhyamik) Mathematics Chapter 11 'Construction of Circumcircle and Incircle of a Triangle'. 20 MCQs, 20-minute timer, instant scorecard, and step-by-step solutions aligned with official board syllabus.

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Instant Scorecard Step-by-Step Solutions WBBSE Madhyamik Blueprint

Chapter Study Guide & In-Depth Notes

Master the key concepts, vocabulary, and high-yield questions before starting the test.

3 Sections

WBBSE Class 10 Mathematics: Theorems related to Circle (Chapter 3)

WBBSE Madhyamik Ganit Prakash Ch 3

In the WBBSE Class 10 Madhyamik Mathematics curriculum (Ganit Prakash / গণিত প্রকাশ), Theorems related to Circle (Chapter 3) forms the foundation of secondary Euclidean geometry (বৃত্ত সম্পর্কিত উপপাদ্য). A circle is defined as the geometric locus of all points in a plane equidistant from a fixed point called the centre ($O$), with the constant distance denoted as the radius ($r$). A straight line segment connecting any two points on the circumference is known as a chord ($AB$). The longest chord passing through the centre is the diameter ($d = 2r$).

The fundamental geometric pillars of this chapter are established by Theorems 32 and 33: "The perpendicular drawn from the centre of a circle to a chord which is not a diameter, bisects the chord." Conversely, "The line segment joining the centre of a circle and the midpoint of a chord (not a diameter) is perpendicular to the chord." If $OM \perp AB$, then $AM = MB = \frac{1}{2}AB$, and inside right-angled triangle $\triangle OMA$, the Pythagorean relation governs all metric dimensions: $$r^2 = d^2 + \left(\frac{L}{2}\right)^2$$ where $r = OA$ is the radius, $d = OM$ is the perpendicular distance from the centre to the chord, and $L = AB$ is the chord length.

Furthermore, chords of a circle equidistant from the centre are equal in length ($d_1 = d_2 \iff L_1 = L_2$), and conversely, equal chords are strictly equidistant from the centre. In concentric circles (circles sharing the same centre with radii $r < R$), a chord of the outer circle tangent to the inner circle is bisected at the point of tangency by the radius of the inner circle. Thorough conceptual and algebraic mastery of these theorems guarantees 100% marks in both geometric riders (proofs) and objective numerical problems in the WBBSE Madhyamik board examination.

Core Moral & Key Takeaway:

Mastering centre-chord perpendicularity $OM \perp AB$, chord bisection $AM = \frac{L}{2}$, and the Pythagorean metric $r^2 = d^2 + (L/2)^2$ guarantees complete success in WBBSE Madhyamik circle geometry.

Core Theorems, Formulas & Geometric Terminology

Theorems & Formula Sheet
Term / Formula Type Definition / Meaning Explanation / Notes Example / Usage
Theorem 32 Theorem Line joining centre to chord midpoint is perpendicular: OM ⊥ AB. Line segment joining circle centre to chord midpoint is perpendicular to the chord. "If M is midpoint of chord AB, then ∠OMA = 90°."
Theorem 33 Theorem Perpendicular from centre bisects chord: AM = MB = L/2. Perpendicular drawn from centre of circle to a chord bisects the chord. "If OM ⊥ AB, then AM = MB = AB/2."
Chord-Radius Pythagorean Metric Master Formula r² = d² + (L/2)² Radius squared equals perpendicular distance squared plus half-chord squared. "If r = 10 cm, d = 6 cm, then half-chord = √(100 - 36) = 8 cm, chord = 16 cm."
Perpendicular Distance (d) Derived Formula d = √(r² - (L/2)²) Perpendicular distance from centre = √(radius² - (chord/2)²). "For r = 13 cm and chord = 24 cm, d = √(169 - 144) = 5 cm."
Chord Length (L) Derived Formula L = 2 · √(r² - d²) Total chord length = 2 × √(radius² - distance²). "For r = 5 cm and d = 3 cm, L = 2 × √(25 - 9) = 8 cm."
Equidistant Chords Property Property L₁ = L₂ ⇔ d₁ = d₂ Equal chords are equidistant from the centre and vice versa. "Two chords of 14 cm each are at identical distances from centre."
Parallel Chords on Opposite Sides Distance Rule Total distance D = d₁ + d₂ Distance between parallel chords on opposite sides of centre is the sum of distances. "If d₁ = 4 cm and d₂ = 3 cm, then distance between them is 4 + 3 = 7 cm."
Concentric Circles Tangent Chord Identity L = 2√(R² - r²) Length of outer chord touching inner circle of radius r. "For R = 5 cm, r = 3 cm, chord length L = 2√(25 - 9) = 8 cm."

Key Features & Devices Explored:

Right-Triangle Construction Method
Always drop perpendicular OM from centre O to chord AB and join radius OA to formulate right-triangle OMA.
Half-Chord Caution Rule
Always apply Pythagorean metric using half-chord length (L/2), and double the result to get full chord length.
Parallel Chords Position Test
Verify whether chords lie on opposite sides (d₁ + d₂) or the same side (|d₁ - d₂|) of the circle centre.
Concentric Circle Tangent Property
Radius to tangent forms 90° angle at contact point, making inner radius r the perpendicular distance d.

High-Yield Revision Points & Solved Madhyamik Board Exam Q&A

WBBSE Exam Revision Notes
⚡ High-Yield Exam Takeaways:
  • Common Pythagorean Triplets: Questions in Madhyamik examinations frequently utilize standard integer triplets: (3, 4, 5), (6, 8, 10), (5, 12, 13), and (8, 15, 17).
  • Chord Bisection: The perpendicular from centre bisects the chord ($AM = MB$). This geometric property is frequently tested in 2-mark short questions.
  • Diameter Property: The diameter is the maximum possible chord ($d = 2r$), and its perpendicular distance from the centre is zero ($d = 0$).
  • Distance vs. Chord Length: Chords closer to the centre are longer; as perpendicular distance increases, chord length decreases.
  • Rider Presentation: When writing geometric proofs for Madhyamik riders, format your answer strictly into four sections: Given, To Prove, Construction, and Proof.
Frequently Asked Questions & Model Answers:
Q1 In a circle of radius 10 cm, find the perpendicular distance from the centre to a chord of length 16 cm.
Answer: Here radius $r = 10\text{ cm}$ and chord $AB = 16\text{ cm}$. Draw perpendicular $OM \perp AB$. By Theorem 33, $OM$ bisects $AB$: $$AM = \frac{1}{2}AB = \frac{16}{2} = 8\text{ cm}$$ In right-angled $\triangle OMA$, by Pythagoras theorem: $$OM = \sqrt{OA^2 - AM^2} = \sqrt{10^2 - 8^2} = \sqrt{100 - 64} = \sqrt{36} = 6\text{ cm}$$ Hence, the perpendicular distance from the centre is 6 cm.
Q2 In a circle of radius 13 cm, a chord is at a distance of 5 cm from the centre. Find the total length of the chord.
Answer: Given radius $r = 13\text{ cm}$ and distance $d = OM = 5\text{ cm}$. In right $\triangle OMA$: $$AM = \sqrt{r^2 - d^2} = \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12\text{ cm}$$ Since $OM$ bisects the chord $AB$, the total chord length is: $$AB = 2 \times AM = 2 \times 12 = 24\text{ cm}$$
Q3 In a circle of radius 5 cm, two parallel chords of lengths 8 cm and 6 cm are drawn on opposite sides of the centre. Find the distance between the two chords.
Answer: Here radius $r = 5\text{ cm}$. For first chord $AB = 8\text{ cm}$: $$d_1 = \sqrt{5^2 - (8/2)^2} = \sqrt{25 - 16} = \sqrt{9} = 3\text{ cm}$$ For second chord $CD = 6\text{ cm}$: $$d_2 = \sqrt{5^2 - (6/2)^2} = \sqrt{25 - 9} = \sqrt{16} = 4\text{ cm}$$ Since the chords are on opposite sides of the centre, total distance between them is: $$D = d_1 + d_2 = 3 + 4 = 7\text{ cm}$$
Q4 Two concentric circles have radii 5 cm and 3 cm. Find the length of the chord of the outer circle which touches the inner circle.
Answer: Let radius of outer circle $R = 5\text{ cm}$ and inner circle $r = 3\text{ cm}$. The chord $AB$ of the outer circle touches the inner circle at $M$, so $OM \perp AB$ and $OM = r = 3\text{ cm}$. In right $\triangle OMA$: $$AM = \sqrt{R^2 - r^2} = \sqrt{5^2 - 3^2} = \sqrt{25 - 9} = \sqrt{16} = 4\text{ cm}$$ Since $OM$ bisects chord $AB$, total length is: $$AB = 2 \times AM = 2 \times 4 = 8\text{ cm}$$

Sample Questions Preview

Free Sneak Peek

Here is a sneak peek of the questions included in this test set. Test your preparation by viewing the answer and detailed explanation before starting the timed exam.

Q1
The center of the circumcircle (circumcenter) of a triangle is the point of concurrence of:
একটি ত্রিভুজের বৃত্তের কেন্দ্র (বৃত্তকেন্দ্র) হল এর সহমতের বিন্দু:
किसी त्रिभुज के परिवृत्त का केंद्र (परिकेंद्र) किसका समागम बिंदु होता है:
A
Internal bisectors of the anglesকোণের অভ্যন্তরীণ দ্বিখণ্ডকकोणों के आंतरिक समद्विभाजक
B
Perpendicular bisectors of the sides of the triangleত্রিভুজের বাহুর লম্ব দ্বিখণ্ডকत्रिभुज की भुजाओं के लम्ब समद्विभाजक
C
Altitudes of the triangleত্রিভুজের উচ্চতাत्रिभुज की ऊंचाई
D
Medians of the triangleত্রিভুজের মধ্যকत्रिभुज की माध्यिकाएँ
Correct Answer: Option B
Explanation:
The circumcenter is the point of intersection of the perpendicular bisectors of the sides of the triangle.
পরিকেন্দ্র হল ত্রিভুজের বাহুর লম্ব বিভাজকগুলির ছেদ বিন্দু।
परिकेन्द्र त्रिभुज की भुजाओं के लम्ब समद्विभाजक का प्रतिच्छेदन बिंदु है।
Q2
The center of the incircle (incenter) of a triangle is the point of concurrence of:
একটি ত্রিভুজের অন্তর্বৃত্তের কেন্দ্র (ইনসেন্টার) হল এর সহমতের বিন্দু:
किसी त्रिभुज के अंतःवृत्त का केंद्र (अंतःकेन्द्र) किसका समागम बिंदु होता है:
A
Internal bisectors of the three angles of the triangleত্রিভুজের তিনটি কোণের অভ্যন্তরীণ দ্বিখণ্ডকत्रिभुज के तीनों कोणों के आंतरिक समद्विभाजक
B
Mediansমিডিয়ানमाध्यिकाओं
C
Perpendicular bisectors of the sidesবাহুর লম্ব দ্বিখণ্ডকभुजाओं के लम्ब समद्विभाजक
D
Altitudesউচ্চতাऊंचाई
Correct Answer: Option A
Explanation:
The incenter is the point where the angle bisectors of the three interior angles intersect.
ইনসেন্টার হল সেই বিন্দু যেখানে তিনটি অভ্যন্তরীণ কোণের কোণ দ্বিখণ্ডক ছেদ করে।
अंतःकेंद्र वह बिंदु है जहां तीन आंतरिक कोणों के कोण समद्विभाजक प्रतिच्छेद करते हैं।

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Syllabus Concepts Evaluated in this Chapter

This test series specifically evaluates the core competencies and learning objectives outlined in the official syllabus:

  • Definitions and fundamental properties of circle, centre, radius, diameter, chord, and arc.
  • Theorem 32: Formal proof that the line joining the centre to the midpoint of a chord is perpendicular to the chord.
  • Theorem 33: Formal proof and applications of perpendicular from centre bisecting the chord.
  • Application of Pythagorean metric $r^2 = d^2 + (L/2)^2$ to solve numerical board exam problems.
  • Calculating distance between parallel chords lying on opposite sides or the same side of the centre.
  • Determining lengths of tangent chords in systems of concentric circles.

Exam Rules & Test Taking Guidelines

  • Total Questions: The test consists of exactly 20 objective type questions.
  • Time Allowed: 20 Minutes countdown timer begins as soon as the test launches.
  • Marking Scheme: +1.0 mark is awarded for each correct answer.
  • Zero Penalty: There is 0.0 negative marking for incorrect or unattempted questions.
  • Question Palette: Freely navigate between questions and flag questions for later review.
  • Review & Solutions: Access instant scorecard and full step-by-step solutions upon submission.

Frequently Asked Questions (FAQs)

Yes! Foundation Practice Set 01 for every chapter in West Bengal Board (WBBSE) Class 10 (Madhyamik) Mathematics is 100% free to attempt. Students can test their preparation without any subscription or payment.
This test strictly follows the official West Bengal Board (WBBSE) syllabus and curriculum for Class 10 (Madhyamik), covering textbook concepts, formulas, and board-standard MCQs for Chapter 11.
This dedicated test page is conducted in English Medium. If you prefer taking the test in Bengali or Hindi, you can switch using the language switcher bar at the top or toggle languages during the exam.
The test comprises 20 MCQs to be completed in 20 minutes. Each question carries +1.0 mark for a correct response, and there is 0.0 negative marking for incorrect or unattempted questions.
Yes! Instantly after submitting your test, you receive a comprehensive digital scorecard with accuracy analysis, time breakdown, and step-by-step verified explanations for all 20 questions.

Practice Sets in this Chapter

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Set 02 (Advanced)

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Construction of Circumcircle and Incircle of a Triangle (Set 01 (Foundation Level))
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