Quantitative Aptitude

Geometry

1,950 Questions
All Circle Tangents and Chords 772 Coordinate Geometry 360 Circle and Arc Measures 216 Triangle Centers and Medians 175 Composite and Inscribed Shapes 174 Similarity, Congruence, and Proportionality 118 Right Triangles and Pythagorean Theorem 100 Standard 2D Mensuration 22 Polygon and Angle Sums 10 3D Solid Mensuration 3

Geometry Questions

Multiple choice

Let $AC$ be an $Arc$ of circle subtending a right angle at the centre. The point $B$ divides $Arc\ AC$ in ratio $1 : 2$. If $\overline {OA}=\overline {a}$ and $\overline {OB}=\overline {b}$ then, Find $\overline {OC}$ in terms of $\overline {a}$ and $\overline {b}$
  1. $2\overline {b}+\overline {a}\sqrt{3}$
  2. $2\overline {b}-\overline {a}\sqrt{3}$
  3. $-2\overline {b}-\overline {a}\sqrt{3}$
  4. $3\overline {b}+\overline {a}\sqrt{2}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Arc AC subtends 90 degrees. B divides AC in 1:2, so angle AOB = 30 degrees and angle BOC = 60 degrees. Using vector rotation or geometry, OC can be expressed as 2b - a*sqrt(3) based on the geometry of the points on the circle.

Multiple choice

A chord $AB$ of a circle subtends an angle of $50^o$ at a point on the circumference of the circle. The acute angle between the tangents at $A$ and $B$ has magnitude
  1. $80^o$
  2. $65^o$
  3. $75^o$
  4. $85^o$
  5. $82^o$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The angle between tangents at the endpoints of a chord is 180 degrees minus the central angle. The central angle is twice the inscribed angle, so 2 * 50 = 100 degrees. The angle between tangents = 180 - 100 = 80 degrees.

Multiple choice

Angles subtended by chords $AC$ and $BC$ at the centre $O$ of the circle are ${ 55 }^{ o }$ and ${ 155 }^{ o }$ respectively. What is the measure of $\angle ACB$?
  1. ${ 65 }^{ o }$
  2. ${ 75 }^{ o }$
  3. ${ 105 }^{ o }$
  4. ${ 135 }^{ o }$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The angle subtended by a chord at the center is twice the angle at the circumference. However, this question describes angles subtended by chords AC and BC at the center. The angle ACB is half the angle subtended by the arc AB at the center. If the angles subtended by AC and BC are 55 and 155, the angle subtended by arc AB is 360 - (55+155) = 150. Thus, angle ACB = 150/2 = 75 degrees.

Multiple choice

The angle of a sector in a given circle is $40$ degrees and the area of the sector is equal to $20 cm^2$ . Calculate the arc length of the sector.
  1. $6.5$ cm
  2. $7.4$ cm
  3. $5.3$ cm

  4. none of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The area of a sector is given by A = (theta / 360) * pi * r^2, which means 20 = (40 / 360) * pi * r^2, so r = sqrt(180 / pi) approx 7.57 cm. The arc length is L = (theta / 360) * 2 * pi * r = (40 / 360) * 2 * pi * 7.57 approx 5.28 cm, which rounds to 5.3 cm.

Multiple choice

A chord of a circle is equal to its radius. The angle subtended by this chord at a point in major segment is
  1. $ { 60 }^{ o } $
  2. $ { 30 }^{ o } $
  3. $ { 120 }^{ o } $
  4. $ { 90 }^{ o } $
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

A chord equal to the radius forms an equilateral triangle with the center, subtending 60 degrees at the center. By the circle theorem, the angle subtended at the circumference in the major segment is half the angle at the center, which is 60 / 2 = 30 degrees.

Multiple choice

Let AB be a sector of a circle with centre O and radius d. $\angle$AOB $=\theta\left(< \displaystyle\frac{\pi}{2}\right)$, and D be a point on OA such that BD is perpendicular OA. Let E be the midpoint of BD and F be a point on the arc AB such that EF is parallel to OA. Then the ratio of length of the arc AF to the length of the arc AB is?
  1. $\displaystyle\frac{1}{2}$
  2. $\displaystyle\frac{\theta}{2}$
  3. $\displaystyle\frac{1}{2}\sin\theta$
  4. $\displaystyle{\sin^{-1}\left(\displaystyle\frac{1}{2}\sin\theta\right)}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Multiple choice

In a circle of radius $7$ cm, arc $AB$ makes one angle $\displaystyle 90^{\circ}$ at the centre O. Find the length of the major arc $AB$.
  1. $28$ cm
  2. $31$ cm
  3. $33$ cm
  4. $35$ cm
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The circumference of the circle is 2 * pi * r = 2 * pi * 7 = 14 * pi. The minor arc for a 90 degree angle is 1/4 of the circumference, which is 3.5 * pi. The major arc is the remaining 3/4 of the circumference, which is 10.5 * pi, approximately 33 cm.

Multiple choice

Angle ABC is $40^0$ and the area of the circle is 81$\pi$. If CB is a diameter of the circle, how long is arc AXC?
  1. $4\pi$ units
  2. $2\pi$ units
  3. $\pi$ units
  4. $\dfrac {\pi}{2}$ units
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Area = 81pi = pi * r^2, so r = 9. Diameter CB = 18. In triangle ABC, angle B = 180 - 90 - 40 = 50 degrees. The arc length AXC corresponds to the central angle subtended by chord AC. Since CB is diameter, angle CAB = 90. Arc AC = (angle ABC / 180) * pi * r is incorrect; the angle at center is 2 * angle ABC = 80 degrees. Arc length = (80/360) * 2 * pi * 9 = 4pi.

Multiple choice

lf ABCDEF is a regular hexagon inscribed in a unit circle, then the product of lengths of lines AC and AE is

  1. 3
  2. ${\frac{3}{4}}$
  3. $3\sqrt{3}$
  4. $\displaystyle \frac{3\sqrt{3}}{2}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

In a regular hexagon inscribed in a unit circle, the side length is 1. AC is a diagonal spanning two sides, length = 2 * sin(120/2) = sqrt(3). AE is also a diagonal spanning two sides, length = sqrt(3). Product = sqrt(3) * sqrt(3) = 3.