Mathematics · Quantitative Aptitude

Circle and Arc Properties

115 Questions

Circle and arc properties involve calculating arc lengths, understanding radius relationships, and solving geometric proofs. These geometry concepts are crucial for quantitative aptitude tests. Review these questions to improve your spatial reasoning and accuracy.

Arc length calculationsCircle theoremsRadius and diameterInscribed polygonsCentral angles

Circle and Arc Properties Questions

Multiple choice maths constructions mid-point formula midpoints division of a line segment

Find the coordinates of the centre of a circle, if the coordinates of the end points of a diameter being $(-3,8)$ and $(5,6)$.

  1. $(-1,7)$
  2. $(2,7)$
  3. $(-2,7)$
  4. $(1,7)$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The centre of the circle lies at the mid point of the diameter.

Mid point of two points $ { (x } _{ 1 },{ y } _{ 1 }) $ and $ { (x } _{ 2 },{ y } _{

2 }) $ is  calculated by the formula $ \left( \dfrac { { x } _{ 1 }+{ x

} _{ 2 } }{ 2 } ,\dfrac { { y } _{ 1 }+y _{ 2 } }{ 2 }  \right) $
Using this formula, mid point of the diameter $= \left( \dfrac { -3 + 5}{ 2 } ,\dfrac { 8 + 6 }{ 2 }  \right) = (1,7) $

Multiple choice maths constructions mid-point formula midpoints division of a line segment

Find the centre of circle, if the coordinates of two ends of diameter are $(-1, 7)$ and $(11, 5)$

  1. $(5, 6)$
  2. $(-3, 2)$
  3. $(10, 12)$
  4. $(12, 2)$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$x = \dfrac {11 - 1}{2} = 5$ and $y = \left (\dfrac {5 + 7}{2}\right )$ $= 6$ (hint: using mid-point formula for centre).
$\therefore (5, 6)$ is coordinate of centre

Multiple choice maths constructions mid-point formula midpoints division of a line segment

The centre of a circle is at $(-6,4)$. If one end of a diameter of the circle is at the origin, then find the other end.

  1. $(-12,8)$
  2. $(-12,-8)$
  3. $(12,8)$

  4. None of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Take coordinate of other end of diameter as $P(x,y)$.


since, origin divides the diameter of a circle in $2$ equal parts, hence origin will be mid-point of $(0,0)$ and $(x,y)$.


By midpoint theorem:

$x=\dfrac{x _1+x _2}{2}$ and $y=\dfrac{y _1+y _2}{2}$

$=>-6=\dfrac{0+x}{2}=\dfrac{x}{2}$

$=>x=-12$

And,

$=>4=\dfrac{y+0}{2}=\dfrac{y}{2}$

$=>y=8$.

So, $P(x,y)=(-12,8)$

Multiple choice maths when lines join trapeziums and kites quadrilaterals and their properties closed figures

Given that a right angled trapezium has an inscribed circle. then " the length of the right angled leg is the Harmonic mean of the lengths of bases. "that  statement is __

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

For a right-angled trapezium with an inscribed circle, the height (the right-angled leg) is indeed the geometric mean of the bases, and the harmonic mean relationship is a known property.

Multiple choice

In a circle, the distance from the center to any point on the circle is called the:

  1. Radius

  2. Diameter

  3. Circumference
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

In a circle, the distance from the center to any point on the circle is called the radius.

Multiple choice

What is the name of the theorem that states that the diagonals of a cyclic quadrilateral are perpendicular if and only if the opposite sides are equal?

  1. Brahmagupta's theorem

  2. Ptolemy's theorem

  3. Pythagoras' theorem

  4. Euler's theorem
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Brahmagupta's theorem states that the diagonals of a cyclic quadrilateral are perpendicular if and only if the opposite sides are equal.

Multiple choice

What is the name of the theorem that states that the area of a cyclic quadrilateral is equal to the product of its semiperimeter and the radius of its incircle?

  1. Brahmagupta's theorem

  2. Ptolemy's theorem

  3. Pythagoras' theorem

  4. Euler's theorem
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Euler's theorem states that the area of a cyclic quadrilateral is equal to the product of its semiperimeter and the radius of its incircle.

Multiple choice

What is the name of the theorem that states that the area of a cyclic quadrilateral is equal to the square of the radius of its incircle multiplied by the cotangent of half the sum of its opposite angles?

  1. Brahmagupta's theorem

  2. Ptolemy's theorem

  3. Pythagoras' theorem

  4. Euler's theorem
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Euler's theorem states that the area of a cyclic quadrilateral is equal to the square of the radius of its incircle multiplied by the cotangent of half the sum of its opposite angles.

Multiple choice

What does the symbol (\pi) represent?

  1. Euler's number

  2. The golden ratio

  3. The ratio of a circle's circumference to its diameter

  4. The square root of 2
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

(\pi) is a mathematical constant that represents the ratio of a circle's circumference to its diameter. It is an irrational number and is approximately equal to 3.14159.

Multiple choice

What is the name of the mathematical concept that describes the relationship between the circumference and diameter of a circle?

  1. Pythagorean Theorem

  2. Pascal's Triangle

  3. Euler's Formula

  4. Pi
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Pi is the mathematical constant that represents the ratio of a circle's circumference to its diameter.

Multiple choice

What is the name of the mathematical constant that represents the ratio of the circumference of a circle to its diameter?

  1. Pi (π)

  2. Euler's number (e)

  3. Golden ratio (φ)

  4. Avogadro's number (Nₐ)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Pi (π) is a mathematical constant that represents the ratio of the circumference of a circle to its diameter.

Multiple choice

What is the name of the mathematical constant that represents the ratio of a circle's circumference to its diameter?

  1. Pi

  2. Euler's number

  3. Golden ratio

  4. Square root of 2
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Pi is a mathematical constant that represents the ratio of a circle's circumference to its diameter.

Multiple choice

What is the topological dimension of a circle?

  1. 0

  2. 1

  3. 2

  4. 3
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The topological dimension of a space is a topological invariant that characterizes the number of independent directions in which the space can be continuously deformed. A circle is a one-dimensional space because it can be continuously deformed into a line segment.

Multiple choice

What is the homology dimension of a circle?

  1. 0

  2. 1

  3. 2

  4. 3
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The homology dimension of a space is the largest integer n such that the nth homology group of the space is nonzero. The homology dimension of a circle is 1 because the first homology group of a circle is nonzero, while all higher homology groups are zero.

Multiple choice

What is the angle between a great circle and a small circle?

  1. The angle between the two planes that contain the great circle and the small circle

  2. The angle between the two lines that intersect the great circle and the small circle at right angles

  3. The angle between the two lines that are tangent to the great circle and the small circle at the same point

  4. The angle between the two lines that are parallel to the great circle and the small circle at the same point
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The angle between a great circle and a small circle is the angle between the two planes that contain the great circle and the small circle.