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 perimeter of a certain sector of a circle is equal to the length of the arc of the semicircle. Then the angle at the centre of the sector in radians is.
  1. $\pi - 2$
  2. $\pi + 2$
  3. $\dfrac {\pi}{3}$
  4. $\dfrac {2\pi}{3}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The perimeter of a sector is 2r + r*theta, where r is radius and theta is the central angle. The length of a semicircle arc is pi*r. Setting 2r + r*theta = pi*r, we divide by r to get 2 + theta = pi, so theta = pi - 2.

Multiple choice

The radius of the circle, which is touched by the line $y=x$ and has its centre on the positive direction of x-axis and also cuts-off a chord of length $2$ units along the line $\sqrt { 3 } y-x=0$, is
  1. $\sqrt { 5 } $
  2. $\sqrt { 3 } $
  3. $\sqrt { 2 } $
  4. $1$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The center is (h, 0) and the circle touches y=x (x-y=0), so the radius r = |h-0|/sqrt(1^2+(-1)^2) = |h|/sqrt(2). The chord length formula for the line x-sqrt(3)y=0 is 2*sqrt(r^2-d^2)=2, where d is the distance from (h, 0) to the line. Solving these equations yields r = sqrt(2).

Multiple choice

If the orthocentre, centroid, incentre and circumcentre in a $\triangle ABC$ coincide with each other and if the length of side $AB$ is $\sqrt { 75 } $ units, then the length of the altitude of the triangle through the vertex $A$ is
  1. $\sqrt { 3 } $
  2. $3$ units
  3. $\cfrac { \sqrt { 15 } }{ 2 } $ units
  4. $\cfrac { 15 }{ 2 } $
  5. $\cfrac { \sqrt { 5 } }{ 2 } $
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

If orthocentre, centroid, incentre, and circumcentre coincide, the triangle is equilateral. In an equilateral triangle, the altitude is (sqrt(3)/2) * side. Altitude = (sqrt(3)/2) * sqrt(75) = (sqrt(3)/2) * 5 * sqrt(3) = (5 * 3) / 2 = 15/2.

Multiple choice

Let $C_1, C_2$ be two circles touching each other externally at the point A and let AB be the diameter of circle $C_1$. Draw a secant $BA_3$ to circle $C_2$, intersecting circle $C_1$ at a point $A_1(\neq A)$, and circle $C_2$ at points $A_2$ and $A_3$. If $BA_1=2$, $BA_2=3$ and $BA_3=4$, then the radii of circles $C_1$ and $C_2$ are respectively.
  1. $\displaystyle\frac{\sqrt{30}}{5}, \frac{3\sqrt{30}}{10}$
  2. $\displaystyle\frac{\sqrt{5}}{2}, \frac{7\sqrt{5}}{10}$
  3. $\displaystyle\frac{\sqrt{6}}{2}, \frac{\sqrt{6}}{2}$
  4. $\displaystyle\frac{\sqrt{10}}{3}, \frac{17\sqrt{10}}{30}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Multiple choice

Suppose $S_1$ and $S_2$ are two unequal circles; $AB$ and $CD$ are the direct common tangents to these circles. A transverse common tangent $PQ$ cuts $AB$ in $R$ and $CD$ in $S$. If $AB=10$, then $RS$ is:
  1. $8$
  2. $9$
  3. $10$
  4. $11$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

For two circles with direct common tangents AB and CD, and a transverse tangent PQ intersecting them at R and S, the length of the segment RS between the direct tangents is equal to the length of the direct common tangent itself. Thus, RS = AB = 10.

Multiple choice

On the circle with centre O, points A and B are such that OA$=$AB. A point C is located on the tangent at B to the circle such that A and C are on the opposite sides of the lines OB and AB$=$BC. The line segment AC intersects the circle again at F. Then the ratio $\angle$BOF$:\angle$BOC is equal to:
  1. $1:2$
  2. $2:3$
  3. $3:4$
  4. $4:5$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Using geometric properties of the circle and tangent, one can derive the angles. The configuration leads to specific triangle relationships where the ratio of the angles BOF and BOC is 2:3.

Multiple choice

Let ABC be a tringle in which the line joining the circumcentre and incentre is parallel to the base of the triangle, then answer the given question: Then range of $ \angle\ A$ is,
  1. $ \left[ \dfrac { \pi }{ 6 } ,\dfrac { \pi }{ 3 } \right]$
  2. $ \left[ \dfrac { \pi }{ 3 } ,\dfrac { \pi }{ 2 } \right]$
  3. $ \left[ \dfrac { \pi }{ 6 } ,\dfrac { 2\pi }{ 3 } \right] -\left\{ \dfrac { \pi }{ 2 } \right\}$
  4. $ \left\{ 0.\dfrac { \pi }{ 2 } \right\}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

For the line joining the circumcenter and incenter to be parallel to the base, the triangle must satisfy specific geometric properties related to the Euler line and triangle centers. This is a known result in triangle geometry.

Multiple choice

The chord of contact of the tangents drawn from a point on the circle, ${x^2} + {y^2} = {a^2}$ to the circle ${x^2} + {y^2} = {B^2}$ touch the circle ${x^2} + {y^2} = {c^2}$ then a,b,c are in :

  1. A.P.

  2. G.P.

  3. H.P.
  4. $None of these$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let a point on the circle x^2 + y^2 = a^2 be (a cos theta, a sin theta). The chord of contact of this point with respect to the circle x^2 + y^2 = b^2 is given by ax cos theta + ay sin theta = b^2. Since this line touches the circle x^2 + y^2 = c^2, its perpendicular distance from the origin must equal c, which gives b^2 / a = c, or b^2 = ac. This confirms that a, b, and c are in Geometric Progression (G.P.).