Circles - Theorems and Properties - Proof-based Questions - Problem Set MCQ & Objective Questions
Understanding the theorems and properties of circles is crucial for students preparing for school and competitive exams. The "Circles - Theorems and Properties - Proof-based Questions - Problem Set" focuses on essential concepts that frequently appear in exams. Practicing MCQs and objective questions not only enhances your grasp of these topics but also boosts your confidence and scores in examinations.
What You Will Practise Here
Fundamental properties of circles, including radius, diameter, and circumference.
Theorems related to angles in circles, such as inscribed angles and central angles.
Chords, tangents, and secants: definitions and properties.
Proof-based questions involving circle theorems and their applications.
Real-life applications of circle properties in geometry problems.
Diagrams illustrating key concepts for better understanding.
Important formulas related to area and circumference of circles.
Exam Relevance
The topic of circles is a significant part of the mathematics syllabus for CBSE, State Boards, NEET, and JEE. Questions often involve applying theorems to solve problems, proving statements, or calculating values related to circles. Students can expect both direct application questions and proof-based questions that test their conceptual understanding and analytical skills.
Common Mistakes Students Make
Confusing the properties of chords and tangents, leading to incorrect conclusions.
Misapplying theorems, especially in proof-based questions.
Neglecting to draw accurate diagrams, which can result in misunderstandings of the problem.
Overlooking the relationship between angles and arcs in circles.
FAQs
Question: What are the key theorems related to circles that I should focus on? Answer: Key theorems include the angle subtended by a chord, the tangent-secant theorem, and the properties of cyclic quadrilaterals.
Question: How can I improve my performance in proof-based questions on circles? Answer: Practice regularly with various proof-based questions and ensure you understand the underlying concepts and theorems.
Now is the time to enhance your understanding of circles! Dive into our practice MCQs and test your knowledge on "Circles - Theorems and Properties - Proof-based Questions - Problem Set". Master these important questions for exams and achieve your academic goals!
Q. A tangent to a circle is drawn from a point outside the circle. If the distance from the point to the center of the circle is 10 cm and the radius of the circle is 6 cm, what is the length of the tangent?
A.
8 cm
B.
10 cm
C.
12 cm
D.
14 cm
Solution
Using the Pythagorean theorem, the length of the tangent is √(10^2 - 6^2) = √(100 - 36) = √64 = 8 cm.
Q. If two circles intersect at points A and B, and the line segment AB is the common chord, what can be said about the angles subtended by AB at the centers of the circles?
A.
They are equal
B.
They are supplementary
C.
They are complementary
D.
They are unequal
Solution
The angles subtended by the common chord AB at the centers of the circles are equal.
Q. Two triangles are similar if their corresponding angles are equal. If triangle DEF is similar to triangle XYZ, and angle D = 50 degrees, what is the measure of angle X?
A.
50 degrees
B.
60 degrees
C.
70 degrees
D.
80 degrees
Solution
Since the triangles are similar, corresponding angles are equal. Therefore, angle X = angle D = 50 degrees.
Q. What is the measure of the central angle that subtends an arc of length 5 cm in a circle of radius 10 cm?
A.
30 degrees
B.
60 degrees
C.
90 degrees
D.
45 degrees
Solution
The formula for the arc length is L = rθ, where θ is in radians. Thus, θ = L/r = 5/10 = 0.5 radians. Converting to degrees, θ = 0.5 * (180/π) ≈ 28.65 degrees, which is closest to 30 degrees.
