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

Three distinct points A, B and C are given in the 2- dimensional coordinate plane such that the ratio of the distance of any one of them from the point (1, 0) to the distance from the point (-1, 0) is equal to 1/3. Then the circumcentre of the triangle ABC is at the point
  1. $(\frac { 5 }{ 4 } ,0)$
  2. $(\frac { 5 }{ 2 } ,0)$
  3. $(\frac { 5 }{ 3 } ,0)$

  4. (0, 0)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The locus of points whose distances from (1, 0) and (-1, 0) are in ratio 1/3 is a circle (Apollonius circle). The circumcenter of any triangle formed by points on this circle is the center of this circle, which lies on the x-axis at x = 5/4.

Multiple choice

In $\Delta ABC,\,M$ is the mid point of $BC$. Length of $AM$ is $9,\,N$ is a point on $AM$ such that $MN=1$. What is the distance of $N$ from the centroid of the triangle?
  1. $2$
  2. $3$
  3. $5$
  4. $7$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

In a triangle, the centroid G divides the median AM in a 2:1 ratio. AG = (2/3)AM = (2/3)*9 = 6. GM = (1/3)AM = 3. N is on AM such that MN=1. Since M is at the base, G is 3 units from M. N is 1 unit from M. The distance GN = GM - MN = 3 - 1 = 2.

Multiple choice

If $H$ is the orthocentre of a triangle $ABC$, then the radii of the circle circumscribing the triangles $BHC, CHA$ and $AHB$ respectively equal to:
  1. $R, R, R$
  2. $\sqrt { 2 } R,\sqrt { 2 } R,\sqrt { 2 } R$
  3. $2R,2R,2R$
  4. $\cfrac { R }{ 2 } ,\cfrac { R }{ 2 } ,\cfrac { R }{ 2 } $
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The circumradius of a triangle formed by the orthocenter and two vertices is equal to the circumradius of the original triangle ABC. This is a known geometric property.

Multiple choice

$ABC$ is a triangle in which $\angle\ B=90^{o},\ BC=48\ cm$ and $AB=14\ cm$. A circle is inscribed in the triangle, whose centre is $O$. Find radius $r$ of in-circle.
  1. $4\ cm$
  2. $6\ cm$
  3. $7\ cm$
  4. $9\ cm$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

In a right triangle with legs a and b and hypotenuse c, the inradius r = (a + b - c) / 2. Here, a = 14, b = 48, and c = sqrt(14^2 + 48^2) = sqrt(196 + 2304) = sqrt(2500) = 50. Thus, r = (14 + 48 - 50) / 2 = 12 / 2 = 6.

Multiple choice

In a triangle $ABC,\,\angle{A}={30}^{\circ},\,\,H$ is the orthocentre and $D$ is the midpoint of $BC$.Segment $HD$ is produced to $T$ such that $HD=DT$.The ratio of the lengths $\dfrac{AT}{BC}$ is equal to
  1. $1:2$
  2. $2:1$
  3. $3:2$
  4. $2:3$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

In a triangle, the distance from the orthocenter to a vertex is 2R cos A. The segment HD produced to T such that HD=DT makes AT a diameter or related to circumradius. The ratio AT/BC simplifies to 2:1.