2.Same Segment: Angles subtended by the same arc in the same segment are equal.
3.Semicircle: Angle subtended by a diameter at the circumference is 90∘.
Structural Rules
4.Isosceles Triangle: Two radii + one chord = isosceles triangle. Base angles are equal.
5.Cyclic Quadrilateral: Opposite angles sum to 180∘.
Actionable Tip
If stuck, draw the radii to the points defining the angle. This almost always reveals an isosceles triangle or a clear central angle to work with.
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Question 1
Level 1: Warm-up
If an arc subtends an angle of 35∘ at a point on the remaining part of the circumference, what is the angle subtended by the same arc at the center of the circle?
Question 2
Level 1: Warm-up
An arc of a circle subtends an angle of 140∘ at the center. What is the angle subtended by the same arc at any point on the remaining part of the circumference?
Question 3
Level 1: Warm-up
A triangle is inscribed in a circle such that one of its sides is the diameter of the circle. What is the measure of the angle opposite to the diameter?
Question 4
Level 1: Warm-up
If a triangle is inscribed in a circle and one of its interior angles is exactly 90∘, what must be true about the side opposite to this angle?
Question 5
Level 1: Warm-up
Points A, B, and C lie on a circle with center O. If the angle subtended by arc AC at point B on the circumference is 35∘, what is the measure of the central angle ∠AOC in degrees?
Question 6
Level 1: Warm-up
Consider the following statements regarding circle geometry:
Assertion (A): The angle subtended by a diameter at any point on the circumference is 90∘.
Reason (R): The angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circumference.
Question 7
Level 1: Warm-up
A continuous portion of the circumference of a circle is formally known as a:
Question 8
Level 1: Warm-up
Points A,B,C, and D lie on a circle. If ∠ACB=40∘ and both C and D are on the same side of the chord AB, what is the measure of ∠ADB?
Question 9
Level 1: Warm-up
The region bounded by a chord and its corresponding arc in a circle is formally known as a:
Question 10
Level 1: Warm-up
For two angles subtended by the same chord to be exactly equal, what condition must be met regarding their positions?
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2.Same Segment: Angles subtended by the same arc in the same segment are equal.
3.Semicircle: Angle subtended by a diameter at the circumference is 90∘.
Structural Rules
4.Isosceles Triangle: Two radii + one chord = isosceles triangle. Base angles are equal.
5.Cyclic Quadrilateral: Opposite angles sum to 180∘.
Actionable Tip
If stuck, draw the radii to the points defining the angle. This almost always reveals an isosceles triangle or a clear central angle to work with.
Geometry and Mensuration: Solved Questions with Step-by-Step Explanations (10 Problems)
Question 1 · Quantitative AptitudeMCQ
If an arc subtends an angle of 35∘ at a point on the remaining part of the circumference, what is the angle subtended by the same arc at the center of the circle?
A.
35∘
B.
55∘
C.
70∘
D.
105∘
Correct Answer:
C
Step-by-Step Solution
Insight: The central angle is always exactly double the angle at the circumference for the same arc.
Exam route: Multiply the given circumference angle by 2: 35∘×2=70∘.
Learning route: By the Central Angle Theorem, the angle subtended by an arc at the center (∠AOB) is double the angle subtended by the same arc at any point on the remaining circumference (∠APB). Therefore, ∠AOB=2×35∘=70∘.
Answer: C
Question 2 · Quantitative AptitudeMCQ
An arc of a circle subtends an angle of 140∘ at the center. What is the angle subtended by the same arc at any point on the remaining part of the circumference?
A.
70∘
B.
40∘
C.
140∘
D.
280∘
Correct Answer:
A
Step-by-Step Solution
Insight: The angle at the circumference is exactly half the angle at the center for the same arc.
Exam route: Divide the central angle by 2: 140∘/2=70∘.
Learning route: According to the Central Angle Theorem, the angle subtended by an arc at the center of a circle is double the angle subtended by the same arc at any point on the remaining circumference. Therefore, the circumference angle is half of the central angle. Calculation: 140∘/2=70∘.
Answer: A
Question 3 · Quantitative AptitudeMCQ
A triangle is inscribed in a circle such that one of its sides is the diameter of the circle. What is the measure of the angle opposite to the diameter?
A.
45∘
B.
60∘
C.
90∘
D.
180∘
Correct Answer:
C
Step-by-Step Solution
Insight: The angle subtended by a diameter at the circumference is always a right angle.
Exam route: Recall the "angle in a semicircle" theorem: it is always 90∘.
Learning route: A diameter subtends a straight angle (180∘) at the center of the circle. By the Central Angle Theorem, the angle at the circumference is exactly half of the central angle. Therefore, 180∘/2=90∘.
Answer: C
Question 4 · Quantitative AptitudeMCQ
If a triangle is inscribed in a circle and one of its interior angles is exactly 90∘, what must be true about the side opposite to this angle?
A.
It is a radius of the circle
B.
It is a diameter of the circle
C.
It is a tangent to the circle
D.
It is equal to the radius
Correct Answer:
B
Step-by-Step Solution
Insight: The converse of the angle in a semicircle theorem guarantees that a 90∘ inscribed angle subtends a diameter.
Exam route: Recall that an angle of 90∘ at the circumference means the opposite chord passes through the center, making it a diameter.
Learning route: The Angle in a Semicircle theorem states that an angle subtended by a diameter at the circumference is 90∘. The converse is also true: if an inscribed angle is 90∘, the chord opposite to it must be a diameter of the circle, as it subtends a 180∘ angle at the center.
Answer: B
Question 5 · Quantitative AptitudeNAT
Points A, B, and C lie on a circle with center O. If the angle subtended by arc AC at point B on the circumference is 35∘, what is the measure of the central angle ∠AOC in degrees?
Correct Answer:
70.00
Step-by-Step Solution
Key idea: This is a direct application of the Central Angle Theorem.
Step 1: Identify the given angle. The angle at the circumference is ∠ABC=35∘.
Step 2: Apply the theorem. The angle subtended by an arc at the center is double the angle subtended at the remaining part of the circumference.
Step 3: Calculate the central angle. ∠AOC=2×∠ABC=2×35∘=70∘.
Answer: 70.00
Question 6 · Quantitative AptitudeMCQ
Consider the following statements regarding circle geometry:
Assertion (A): The angle subtended by a diameter at any point on the circumference is 90∘.
Reason (R): The angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circumference.
A.
Both A and R are true and R is the correct explanation of A
B.
Both A and R are true but R is NOT the correct explanation of A
C.
A is true but R is false
D.
A is false but R is true
Correct Answer:
A
Step-by-Step Solution
Key idea: This is an assertion-reason question testing the foundational theorems of circle geometry.
Step 1: Evaluate Assertion (A). A diameter subtends a 180∘ angle at the center (a straight line). By the Central Angle Theorem, the angle at the circumference is half of 180∘, which is 90∘. So, A is true.
Step 2: Evaluate Reason (R). This is the exact statement of the Central Angle Theorem. So, R is true.
Step 3: Check the relationship. The truth of A is directly derived from R. Therefore, R is the correct explanation of A.
Answer: A
Question 7 · Quantitative AptitudeMCQ
A continuous portion of the circumference of a circle is formally known as a:
A.
Chord
B.
Arc
C.
Segment
D.
Sector
Correct Answer:
B
Step-by-Step Solution
Insight: This is a direct terminology recall question about the parts of a circle.
Exam route: Match the definition "continuous portion of the circumference" to the term "Arc".
Learning route: A chord is a line segment connecting two points. An arc is the curved portion of the circumference between those two points. A segment is the region bounded by a chord and an arc. A sector is the region bounded by two radii and an arc.
Answer: B
Question 8 · Quantitative AptitudeMCQ
Points A,B,C, and D lie on a circle. If ∠ACB=40∘ and both C and D are on the same side of the chord AB, what is the measure of ∠ADB?
A.
40∘
B.
50∘
C.
80∘
D.
140∘
Correct Answer:
A
Step-by-Step Solution
Insight: Angles subtended by the same arc in the same segment are equal.
Exam route: Since C and D are on the same side of chord AB, ∠ADB=∠ACB=40∘.
Learning route: Both ∠ACB and ∠ADB are subtended by the same arc AB and lie in the same segment (same side of the chord). By the Angles in the Same Segment theorem, these angles must be exactly equal. Therefore, ∠ADB=40∘.
Answer: A
Question 9 · Quantitative AptitudeMCQ
The region bounded by a chord and its corresponding arc in a circle is formally known as a:
A.
Sector
B.
Tangent
C.
Segment
D.
Secant
Correct Answer:
C
Step-by-Step Solution
Insight: This is a direct definition recall of circle parts.
Exam route: Match "region bounded by a chord and an arc" to the term "Segment".
Learning route: A circle is divided into specific regions by its lines. A "sector" is the region bounded by two radii and an arc. A "segment" is the region bounded by a chord and an arc. A "tangent" is a line touching the circle at one point, and a "secant" is a line intersecting the circle at two points.
Answer: C
Question 10 · Quantitative AptitudeMCQ
For two angles subtended by the same chord to be exactly equal, what condition must be met regarding their positions?
A.
They must be on opposite sides of the chord
B.
One must be at the center of the circle
C.
They must be supplementary
D.
They must be on the same side of the chord
Correct Answer:
D
Step-by-Step Solution
Insight: Angles subtended by the same chord are equal only if they are in the same segment.
Exam route: Recall that "same segment" means the points must be on the same side of the chord.
Learning route: The Angles in the Same Segment theorem states that angles subtended by the same arc or chord in the same segment are equal. The "same segment" condition strictly requires the vertices of the angles to be on the same side of the chord. If they are on opposite sides, the angles are supplementary, not equal.