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

If the perimeter of a circle is equal to the perimeter of a regular polygon of 'n' sides, then their areas are in the ratio:
  1. $\displaystyle \tan \left ( \frac{\pi }{n} \right ):\frac{\pi }{n}$
  2. $\displaystyle \cos \left ( \frac{\pi }{n} \right ):\frac{\pi }{n}$
  3. $\displaystyle \sin \left ( \frac{\pi }{n} \right ):\frac{\pi }{n}$
  4. $\displaystyle \cot \left ( \frac{\pi }{n} \right ):\frac{\pi }{n}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

For a circle and a regular n-sided polygon with equal perimeter P, the radius of the circle is r = P/(2*pi). The apothem of the polygon is a = P/(2*n*tan(pi/n)). The ratio of areas is (pi * r^2) / (n * s * a / 2) = (pi * (P/2pi)^2) / (P * a / 2) = tan(pi/n) / (pi/n).

Multiple choice

The length of the common chord of the circles: $x^{2}+y^{2}+6x+5=0$ and $x^{2}+y^{2}+4y-5=0$ is
  1. $\sqrt{\dfrac{12}{13}}$
  2. $\displaystyle \frac{12}{\sqrt{13}}$
  3. $\displaystyle \frac{\sqrt{12}}{13}$
  4. $\sqrt{\dfrac{13}{12}}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Common chord equation: (x^2+y^2+6x+5) - (x^2+y^2+4y-5) = 0 => 6x - 4y + 10 = 0 => 3x - 2y + 5 = 0. Distance from center of first circle (-3, 0) to line: |3(-3) - 2(0) + 5| / sqrt(3^2 + (-2)^2) = |-4| / sqrt(13) = 4/sqrt(13). Radius of first circle = sqrt(3^2 + 0^2 - 5) = 2. Half-length of chord = sqrt(r^2 - d^2) = sqrt(4 - 16/13) = sqrt(36/13) = 6/sqrt(13). Total length = 12/sqrt(13).

Multiple choice

Two concentric circles of radii a and b, where a > b, are given. The length of the chord of the larger circle which touches the smaller circle is :
  1. $\sqrt{a^{2}-b^{2}}$
  2. $\sqrt{a^{2}+b^{2}}$
  3. $2\sqrt{a^{2}-b^{2}}$
  4. $2\sqrt{a^{2}+b^{2}}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The chord of the larger circle tangent to the smaller circle forms a right-angled triangle with the radius of the smaller circle (b) and the radius of the larger circle (a). By Pythagoras theorem, half the chord length is sqrt(a^2 - b^2), so the full length is 2*sqrt(a^2 - b^2).

Multiple choice

The common chord of the circles $x^{2} + y^{2} - 4x - 4y = 0$ and $2x^{2} + 2y^{2} = 32$ subtends at the origin an angle equal to
  1. $\dfrac {\pi}{3}$
  2. $\dfrac {\pi}{4}$
  3. $\dfrac {\pi}{6}$
  4. $\dfrac {\pi}{2}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Circle 1: x^2 + y^2 - 4x - 4y = 0. Circle 2: x^2 + y^2 = 16. Subtracting gives the common chord: -4x - 4y = -16, or x + y = 4. The chord is at distance 2*sqrt(2) from the origin. The angle subtended at the origin is 90 degrees.

Multiple choice

lf the triangle is not a right angled triangle and tangent to circum circle at A and B intersects at F $(z_{6})$ then
  1. $\displaystyle \mathrm{z}_{6}=-\frac{\mathrm{z}_{1}\mathrm{z}_{3}}{\mathrm{z}_{2}}$
  2. $\displaystyle \mathrm{z}_{6}=-\frac{\mathrm{z}_{2}\mathrm{z}_{3}}{\mathrm{z}_{1}}$
  3. $\mathrm{z}_{6}^{2}=\mathrm{z}_{2}\mathrm{z}_{3}$
  4. $\displaystyle \mathrm{z}_{6}=\frac{2\mathrm{z}_{1}\mathrm{z}_{2}}{\mathrm{z}_{1}+\mathrm{z}_{2}}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Multiple choice

$\Delta ABC$ is inscribed in a circle with center $O$. If $\angle OAB = { 40 }^{\circ}$, then $\angle ACB$ is equal to
  1. ${ 50 }^{\circ} $
  2. ${ 40 }^{\circ} $
  3. ${ 60 }^{\circ} $
  4. ${ 70 }^{\circ} $
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

In triangle OAB, OA = OB (radii), so angle OBA = angle OAB = 40 degrees. Angle AOB = 180 - (40 + 40) = 100 degrees. The angle subtended by arc AB at the center is 100 degrees. The angle at the circumference (angle ACB) is half of the angle at the center, so 100 / 2 = 50 degrees.

Multiple choice

The points $A,\,B\;and\;C$ be on a circle in such a way that the $\angle ABC=52^{\circ}$ and $\angle ACB=78^{\circ}$. The measure of the angle subtended at the centre by the arc $BC$ will be
  1. $26^{\circ}$
  2. $50^{\circ}$
  3. $100^{\circ}$
  4. $115^{\circ}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

In triangle ABC, angle A = 180 - (52 + 78) = 50 degrees. The angle subtended by arc BC at the center is twice the angle subtended at the circumference (angle A). Angle = 2 * 50 = 100 degrees.

Multiple choice

$AB$, $BC$ and $CD$ are the three consecutive sides of a regular polygon. If $\angle BAC\, =\, 15^{\circ}$, find the measure of each exterior angle of the polygon.
  1. $20^{\circ}$
  2. $30^{\circ}$
  3. $40^{\circ}$
  4. $70^{\circ}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

In a regular polygon, the triangle formed by the center and two adjacent vertices is isosceles. The angle at the center is 360/n. The base angles are (180 - 360/n)/2 = 90 - 180/n. Given the angle BAC is 15, and using properties of regular polygons, n=12. Exterior angle = 360/12 = 30 degrees.