Understanding "Circles - Tangents & Chords" is crucial for students preparing for school exams and competitive tests. This topic not only forms a significant part of the mathematics syllabus but also enhances problem-solving skills. Practicing MCQs and objective questions on this subject helps in reinforcing concepts and boosts confidence, ultimately leading to better scores in exams.
What You Will Practise Here
Definitions and properties of circles, tangents, and chords
Key theorems related to tangents and chords
Formulas for calculating lengths and angles involving tangents and chords
Diagrams illustrating the relationships between circles, tangents, and chords
Problem-solving techniques for objective questions
Real-life applications of circles and their properties
Sample MCQs with detailed explanations
Exam Relevance
The topic of "Circles - Tangents & Chords" is frequently tested in CBSE, State Boards, NEET, and JEE exams. Students can expect questions that require them to apply theorems, solve for unknown lengths, or analyze diagrams. Common question patterns include multiple-choice questions that assess both conceptual understanding and practical application of theorems.
Common Mistakes Students Make
Confusing the properties of tangents with those of secants
Misapplying theorems related to angles formed by tangents and chords
Overlooking the importance of accurate diagram interpretation
Failing to remember key formulas during problem-solving
FAQs
Question: What is the relationship between a tangent and a radius at the point of contact? Answer: A tangent to a circle is perpendicular to the radius drawn to the point of contact.
Question: How do you find the length of a tangent from an external point to a circle? Answer: The length of the tangent can be calculated using the formula: \( \sqrt{d^2 - r^2} \), where \( d \) is the distance from the external point to the center of the circle and \( r \) is the radius.
Now is the perfect time to enhance your understanding of "Circles - Tangents & Chords". Dive into our practice MCQs and test your knowledge to excel in your exams!
Q. A circle has a radius of 10 cm. What is the length of a chord that is 8 cm away from the center?
A.
12
B.
16
C.
18
D.
20
Solution
Using the formula: chord length = 2√(radius² - distance from center²). Here, chord length = 2√(10² - 8²) = 2√(100 - 64) = 2√36 = 12.
Q. If a chord of a circle is 16 cm long and is 6 cm away from the center, what is the radius of the circle?
A.
10
B.
12
C.
14
D.
16
Solution
Using the formula: radius² = (distance from center to chord)² + (half of chord length)². Here, radius² = 6² + (16/2)² = 36 + 64 = 100, so radius = √100 = 10.
Q. If a tangent and a chord intersect at a point on the circle, and the lengths of the tangent and chord are 6 cm and 8 cm respectively, what is the angle between them?
A.
30 degrees
B.
45 degrees
C.
60 degrees
D.
90 degrees
Solution
The angle between a tangent and a chord at the point of contact is always 90 degrees.
Q. If a tangent and a chord intersect at a point on the circle, what is the relationship between the angle formed and the angle subtended by the chord at the center?
A.
They are equal
B.
They are supplementary
C.
They are complementary
D.
They are unrelated
Solution
The angle formed between the tangent and the chord is equal to the angle subtended by the chord at the center of the circle.
Q. If a tangent to a circle makes a 30-degree angle with a chord drawn to the point of tangency, what is the measure of the angle between the chord and the radius at the point of tangency?
A.
30 degrees
B.
60 degrees
C.
90 degrees
D.
120 degrees
Solution
The angle between the chord and the radius at the point of tangency is equal to the angle between the tangent and the chord, which is 30 degrees. Therefore, the angle between the chord and the radius is 90 - 30 = 60 degrees.
Q. If two chords intersect inside a circle and the lengths of the segments are 3 cm and 4 cm for one chord, and 2 cm and x cm for the other, what is the value of x?
A.
5
B.
6
C.
7
D.
8
Solution
Using the intersecting chords theorem: 3 * 4 = 2 * x, so 12 = 2x, thus x = 6.
Q. If two chords intersect inside a circle, and the lengths of the segments of one chord are 4 cm and 6 cm, what is the length of the other chord if its segments are x cm and y cm?
A.
10
B.
12
C.
14
D.
16
Solution
Using the intersecting chords theorem: 4 * 6 = x * y. If x + y = 10, then x = 4 and y = 6.
Q. If two tangents are drawn from a point outside a circle, and the lengths of the tangents are 7 cm and 7 cm, what is the distance from the point to the center of the circle?
A.
7√2
B.
7
C.
14
D.
10
Solution
The distance from the point to the center is equal to the length of the tangent divided by cos(45°), which is 7√2.
Q. If two tangents are drawn from a point outside a circle, and the lengths of the tangents are 7 cm each, what is the distance from the point to the center of the circle?
A.
7√2
B.
7
C.
14
D.
√49
Solution
The distance from the point to the center is given by the formula: distance = √(tangent length² + radius²). Here, radius = 7, so distance = √(7² + 7²) = √(49 + 49) = √98 = 7√2.
Q. In a circle, if a chord is 12 cm long and the distance from the center to the chord is 5 cm, what is the radius of the circle?
A.
10
B.
12
C.
13
D.
15
Solution
Using the formula: radius² = (distance from center to chord)² + (half of chord length)². Here, radius² = 5² + (12/2)² = 25 + 36 = 61, so radius = √61, which is approximately 7.81.
Q. In a circle, if a chord is 12 units long and the distance from the center to the chord is 5 units, what is the radius of the circle?
A.
10
B.
12
C.
13
D.
15
Solution
Using the formula: radius² = (distance from center to chord)² + (half of chord length)². Thus, radius² = 5² + (12/2)² = 25 + 36 = 61, so radius = √61 ≈ 7.81.
Q. In a circle, if a tangent and a chord intersect at a point on the circle, and the angle between them is 30°, what is the angle subtended by the chord at the center?
A.
30°
B.
60°
C.
90°
D.
120°
Solution
The angle subtended by the chord at the center is twice the angle between the tangent and the chord, so it is 2 * 30° = 60°.
Q. In a circle, if the angle subtended by a chord at the center is 80 degrees, what is the angle subtended at any point on the remaining part of the circle?
A.
40
B.
80
C.
100
D.
160
Solution
The angle subtended at the circumference is half of that at the center, so it is 80/2 = 40 degrees.
