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

In a circle with center $O$, a chord $PQ$ is such that $OM\pm PQ$ meeting $PQ$ at $M$. Then
  1. ${ OQ }^{ 2 }={ OM }^{ 2 }+\dfrac { 1 }{ 2 } { PQ }^{ 2 }$
  2. ${ OQ }^{ 2 }={ OM }^{ 2 }+\dfrac { 1 }{ 4 } { PQ }^{ 2 }$
  3. ${ MQ }^{ 2 }={ OM }^{ 2 }- { OQ }^{ 2 }$
  4. ${ OM }^{ 2 }={ MQ }^{ 2 }- { OQ }^{ 2 }$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

In right triangle OMQ, OQ^2 = OM^2 + MQ^2. Since M is the midpoint of chord PQ, MQ = PQ/2. Thus MQ^2 = (PQ/2)^2 = PQ^2/4. Substituting gives OQ^2 = OM^2 + PQ^2/4.

Multiple choice

From a point $A$, the length of a tangent to a circle is $24$ cm and the distance of $A$ from the center is $25$ cm. The radius of the circle is
  1. $7$ cm
  2. $12$ cm
  3. $15$ cm
  4. $24.5$ cm
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The radius of a circle is perpendicular to the tangent at the point of contact, forming a right-angled triangle. Using the Pythagorean theorem, the radius r satisfies r^2 + 24^2 = 25^2, which simplifies to r^2 = 49, giving a radius of 7 cm.

Multiple choice

In a circle of diameter 10 cm, the length of each of 2 equal and parallel chords is 8 cm, then the distance between these two chords is

  1. 4 cm

  2. 5 cm

  3. 6 cm

  4. 7 cm
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The distance from the center to a chord of length 8 in a circle of radius 5 is sqrt(5^2 - 4^2) = 3. If chords are on opposite sides of the center, distance is 3 + 3 = 6. If on the same side, distance is 3 - 3 = 0. Given the options, 6 is the intended answer.

Multiple choice

Two perpendicular chords intersect in a circle. The segments of one chord are 3 and 4; the segments of the other are 6 and 2. Then the diameter of the circle is:
  1. $\sqrt{89}$
  2. $\sqrt{56}$
  3. $\sqrt{65}$
  4. $\sqrt{75}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

For perpendicular chords intersecting at P, if segments are a, b and c, d, the radius R is given by R^2 = (a^2+b^2+c^2+d^2)/4. R^2 = (3^2+4^2+6^2+2^2)/4 = (9+16+36+4)/4 = 65/4. Diameter = 2R = 2 * sqrt(65/4) = sqrt(65).

Multiple choice

Two chords $AB$ and $CD$ of circle whose centre is $O$, meet at the point $P$ and $\angle AOC=50^{o}, \angle BOD=40^{o}$. Then the value of $\angle BPD$ is:
  1. $60^{o}$
  2. $40^{o}$
  3. $45^{o}$
  4. $75^{o}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The angle subtended by an arc at the center is double the angle subtended at the circumference. Angle ACB = 1/2 * angle AOB = 1/2 * (50 + 40) = 45 degrees. In triangle PBC, the exterior angle BPD = angle PCB + angle PBC, but it is simpler to note that angle BPD = 1/2 * (angle AOC + angle BOD) = 1/2 * (50 + 40) = 45 degrees.