Mathematics · Quantitative Aptitude

Circle and Arc Properties

117 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 tangents and intersecting chords other theorems related to circles touching circles angle made by a chord and a tangent

Opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the center of the circle. 

  1. True

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

For a quadrilateral circumscribing a circle, the triangles formed by the center and the sides have properties such that the central angles subtended by opposite sides sum to 180 degrees.

Multiple choice maths tangents and intersecting chords other theorems related to circles touching circles angle made by a chord and a tangent

Let $C _1$ and $C _2$ be two non concentric circles with $C _2$ lying inside $C _1$. A circle C lying inside $C _1$ touches $C _1$ internally and $C _2$ externally. The locus of the centre of the circle C is :

  1. Ellipse

  2. Circle

  3. Parabola

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

Let the radii of C1 and C2 be R and r, and their centers be O1 and O2. If a circle with center (x, y) and radius r' touches C1 internally and C2 externally, the distance to the centers relates to the radii, resulting in the definition of an ellipse.

Multiple choice maths tangents and intersecting chords other theorems related to circles touching circles angle made by a chord and a tangent

Consider a circle, $x^{2}+y^{2}=1$ and point $P\left(1,\sqrt{3}\right).PAB$ is secant drawn from $P$ intersecting circle in $A$ and $B$ (distinct) then range of $\left|PA\right|+\left|PB\right|$is 

  1. $\left[2\sqrt{3},4\right]$
  2. $\left(2\sqrt{3},4\right]$
  3. $\left(0,4\right]$
  4. $\left(0,2\sqrt{3}\right)$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Point P(1, sqrt(3)) lies on the circle x^2 + y^2 = 4 (radius 2). The secant PAB intersects the circle. The length PA + PB represents the sum of chord segments. The range is determined by the positions of the secant line.

Multiple choice maths tangents and intersecting chords other theorems related to circles touching circles angle made by a chord and a tangent

Two secants PAB and PCD are drawn to a circle from an outside point P. Then, which of the following is true?

  1. PA. PB =PC +CD

  2. PA. PB =PC. PD

  3. PA+PB=PC+PD

  4. PA-PB = PC. CD
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

By the power of a point theorem, for two secants PAB and PCD drawn from an external point P to a circle, the product of the segments of the secants is equal: PA * PB = PC * PD.

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 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.