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 one of the diameters of the circle ${ x }^{ 2 }+{ y }^{ 2 }-2x-6y+6=0$ is a chord to the circle with centre (2, 1), then the radius of the circle is

  1. 3

  2. 2

  3. 3/2

  4. 1
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The circle equation x^2 + y^2 - 2x - 6y + 6 = 0 has center (1, 3) and radius sqrt(1^2 + 3^2 - 6) = 2. A diameter of this circle is a chord of another circle with center (2, 1). The distance from (2, 1) to the line containing the diameter (which passes through (1, 3)) is needed. However, the diameter itself is a chord of the second circle, so the radius of the second circle R = sqrt(d^2 + r^2), where r=2 and d is the distance from (2, 1) to the center (1, 3). d = sqrt((2-1)^2 + (1-3)^2) = sqrt(1+4) = sqrt(5). R = sqrt(5 + 4) = 3.

Multiple choice

The radius of a circle is $2.5$ $cm .$ $AB$ and $CD$ are two parallel chords $2.7$ $cm$ apart. If $A B = 4.8 \mathrm { cm }$ .then $\mathrm { CD }$ is equal to
  1. $4.8$ $cm$
  2. $2.4$ $cm$
  3. $3$ $cm$
  4. $4$ $cm$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let the center be O. Distance to AB is 2.7. Half of AB is 2.4. Radius is 2.5. Distance from center to AB is sqrt(2.5^2 - 2.4^2) = 0.7. Since chords are parallel and 2.7 apart, the distance from center to CD is 2.7 - 0.7 = 2.0. Half of CD = sqrt(2.5^2 - 2.0^2) = 1.5. CD = 3.0 cm.

Multiple choice

If a chord of the circle ${ x }^{ 2 }+{ y }^{ 2 }=8$ makes equal intercepts of length $a$ on the coordinate axes, then
  1. $\left| a \right| <8$
  2. $\left| a \right| <4\sqrt { 2 } $
  3. $\left| a \right| <4$
  4. $\left| a \right| >4$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

A chord making equal intercepts 'a' on axes means the line equation is x/a + y/a = 1, or x + y = a. The distance from the origin (0,0) to the line x + y - a = 0 must be less than the radius sqrt(8). Distance = |a| / sqrt(1^2 + 1^2) = |a| / sqrt(2). So, |a| / sqrt(2) < sqrt(8) implies |a| < sqrt(16) = 4.

Multiple choice

Quadrilateral ABCD is inscribed in a circle with radius $1$ unit AC is the diameter of the circle and $BD=AB$. The diagonal cut at P. If $PC=\dfrac{2}{5}$ then the length of CD is equal to
  1. $\dfrac{2}{3}$
  2. $\dfrac{2}{7}$
  3. $\dfrac{1}{8}$
  4. $\dfrac{3}{4}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Since AC is diameter, angle ABC = 90 degrees. In triangle ABC, BD=AB, so triangle ABD is isosceles. Using geometric properties of chords and segments in a circle, the segments satisfy power of a point or similar triangles. Given PC=2/5 and radius=1, the calculation leads to CD=2/3.

Multiple choice

The value of $c$ for which the quadrilateral formed by the common chord of the circles $x^{2}+y^{2}+2x+2y=0$ and $x^{2}+y^{2}+6x+4y+c=0$ and the line segment joining their centre as diagonals is a rhombus is
  1. $11$
  2. $7$
  3. $9$
  4. $2$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The common chord equation is (x^2+y^2+6x+4y+c) - (x^2+y^2+2x+2y) = 0, which is 4x + 2y + c = 0. For the quadrilateral to be a rhombus, the diagonals must bisect each other at 90 degrees. This condition leads to c = 11.

Multiple choice

The equation of the chord of the circle $x ^ { 2 } + y ^ { 2 } - 6 x + 8 y = 0$ which is bisected at the point $( 5 , - 3 ) ,$ is
  1. $2 x + y - 7 = 0$
  2. $x + 2 y + 1 = 0$
  3. $2 x - y - 13 = 0$
  4. $x - 2 y - 11 = 0$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The equation of a chord bisected at (x1, y1) is T = S1. T = x*x1 + y*y1 - 3(x+x1) + 4(y+y1). S1 = x1^2 + y1^2 - 6*x1 + 8*y1. Substituting (5, -3) gives 5x - 3y - 3(x+5) + 4(y-3) = 25 + 9 - 30 - 24. 2x + y - 27 = -20. 2x + y - 7 = 0.