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

The circumcentre of the triangle formed by $A(1, 2), B(-2, 2), C(1, 5)$ is
  1. $(1, 2)$
  2. $(-2, 2)$
  3. $(1, 5)$
  4. $\left(\dfrac{-1}{2}, \dfrac{7}{2}\right)$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Circumcentre is equidistant from vertices. Distance squared from (x, y) to (1, 2), (-2, 2), (1, 5). (x-1)^2 + (y-2)^2 = (x+2)^2 + (y-2)^2 => x^2-2x+1 = x^2+4x+4 => 6x = -3 => x = -1/2. (x-1)^2 + (y-2)^2 = (x-1)^2 + (y-5)^2 => y^2-4y+4 = y^2-10y+25 => 6y = 21 => y = 7/2.

Multiple choice

If in a triangle $A = ( 1,10 ) ,$ circumcentre $= \left( - \frac { 1 } { 3 } , \frac { 2 } { 3 } \right)$ and orhocentre $= \left( \frac { 11 } { 3 } , \frac { 4 } { 3 } \right)$ then the co-ordinate of mid-point of side opposite to $A$ is
  1. $( 1,6 )$
  2. $( 1,5 )$
  3. $( 1 , - 3 )$
  4. $( 1 , - \dfrac { 11 } { 3 })$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

In any triangle, the centroid G divides the segment joining the orthocenter H and circumcenter O in the ratio 2:1. G = (2O + H)/3 = (2(-1/3, 2/3) + (11/3, 4/3))/3 = ((-2/3+11/3)/3, (4/3+4/3)/3) = (3/3, 8/9) = (1, 8/9). The midpoint M of the side opposite to A satisfies (A + 2M)/3 = G. (1, 10) + 2M = 3(1, 8/9) = (3, 8/3). 2M = (2, -22/3), so M = (1, -11/3).

Multiple choice

Find the circumcentre of the triangle whose vertices are $(0, 0)$, $\left ( 3, \sqrt{3} \right )$ and $\left ( 0, 2\sqrt{3} \right )$.
  1. $\left ( 1, \sqrt{3} \right )$
  2. $\left ( -1, \sqrt{3} \right )$
  3. $\left ( 0, \sqrt{3} \right )$
  4. $\left ( 1, 2\sqrt{3} \right )$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The circumcenter is equidistant from (0,0), (3, sqrt(3)), and (0, 2*sqrt(3)). Let the center be (x,y). x^2 + y^2 = (x-3)^2 + (y-sqrt(3))^2 and x^2 + y^2 = x^2 + (y-2*sqrt(3))^2. From the second, y^2 = y^2 - 4*sqrt(3)*y + 12, so 4*sqrt(3)*y = 12, y = sqrt(3). Substituting into the first, x^2 + 3 = (x-3)^2 + 0, x^2 + 3 = x^2 - 6x + 9, 6x = 6, x = 1.

Multiple choice

The locus of the mid points of the chords passing through a fixed point $( \alpha , \beta )$ of the hyperbola, $\dfrac { x ^ { 2 } } { a ^ { 2 } } - \dfrac { y ^ { 2 } } { b ^ { 2 } } = 1$ is
  1. a circle with centre $\left( \dfrac { \alpha } { 2 } , \dfrac { \beta } { 2 } \right)$
  2. an ellipse with centre $\left( \dfrac { \alpha } { 2 } , \dfrac { \beta } { 2 } \right)$
  3. a hyperbola with centre $\left( \dfrac { \alpha } { 2 } , \dfrac { \beta } { 2 } \right)$
  4. straight line through $\left( \dfrac { \alpha } { 2 } , \dfrac { \beta } { 2 } \right)$
Reveal answer Fill a bubble to check yourself
A Correct answer
Multiple choice

AB is the diameter of a circle inscribed in a regular hexagon. CD is the diameter of the circumcircle of the same hexagon. If the area of the hexagon is 2019, then AB x CD is

  1. 2692

  2. 1346/2

  3. 2019

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

For a regular hexagon with side a, area = (3 * sqrt(3) / 2) * a^2 = 2019. The inscribed circle diameter AB = sqrt(3) * a. The circumcircle diameter CD = 2a. AB * CD = (sqrt(3) * a) * (2a) = 2 * sqrt(3) * a^2. Since (3 * sqrt(3) / 2) * a^2 = 2019, a^2 = 4038 / (3 * sqrt(3)) = 1346 / sqrt(3). AB * CD = 2 * sqrt(3) * (1346 / sqrt(3)) = 2692.

Multiple choice

In a triangle ABC, CD is the bisector of the angle C. If $\displaystyle \cos \frac{C}{2}$ has the value $\displaystyle \frac{1}{3}$ and length of $ CD=6$, then$\displaystyle \left ( \frac{1}{a}+\frac{1}{b} \right )$ has the value equal to
  1. $\displaystyle \frac{1}{9}$
  2. $\displaystyle \frac{1}{12}$
  3. $\displaystyle \frac{1}{6}$

  4. none
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Using the formula for the length of the angle bisector CD = (2ab cos(C/2)) / (a+b). Given CD=6 and cos(C/2)=1/3, we have 6 = (2ab * 1/3) / (a+b). This simplifies to 6 = (2ab) / (3(a+b)), so 18 = 2ab / (a+b), which means (a+b)/ab = 2/18 = 1/9. Thus 1/b + 1/a = 1/9.