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 area of the circle and the area of an regular polygon of sides of perimeter equal to that of the circle are in the ratio of
  1. $\tan { \left( \cfrac { \pi }{ n } \right) } :\cfrac { \pi }{ n } $
  2. $\cot { \left( \cfrac { \pi }{ n } \right) } :\cfrac { \pi }{ n } $
  3. $\cos{ \left( \cfrac { \pi }{ n } \right) } :\cfrac { \pi }{ n } $
  4. $\sin { \left( \cfrac { \pi }{ n } \right) } :\cfrac { \pi }{ n } $
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

For a circle of radius r, area is pi*r^2 and perimeter is 2*pi*r. For a regular polygon with n sides of total perimeter 2*pi*r, each side length is 2*pi*r/n. The area of such a polygon is (n * s^2) / (4 * tan(pi/n)), where s is the side length. Substituting s, the ratio simplifies to tan(pi/n) / (pi/n).

Multiple choice

Let the tangents drawn from the origin to the circle, $x^2 + y^2 - 8x - 4y + 16 = 0$ touch it at the points $A $ and $B$. The $(AB)^2$ is equal to :
  1. $\dfrac{32}{5}$
  2. $\dfrac{56}{5}$
  3. $\dfrac{52}{5}$
  4. $\dfrac{64}{5}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The length of the chord of contact AB is given by 2*r*d / sqrt(r^2 + d^2), where r is the radius and d is the distance from the origin to the center. Here center is (4,2), r=2, d=sqrt(20). Calculation leads to 64/5.

Multiple choice

If the angle of intersection at a point where the two circles with radii $5$cm and $12$cm intersect is $90^o$, then the length (in cm) of their common chord is?
  1. $\dfrac{60}{13}$
  2. $\dfrac{120}{13}$
  3. $\dfrac{13}{2}$
  4. $\dfrac{13}{5}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

For two circles with radii r1=5 and r2=12 intersecting at 90 degrees, the distance between centers d = sqrt(5^2 + 12^2) = 13. The length of the common chord is (2 * r1 * r2) / d = (2 * 5 * 12) / 13 = 120 / 13.

Multiple choice

The tangent to the circle $C_{1} : x^{2} + y^{2} - 2x - 1 = 0$ at the point $(2, 1)$ cuts off a chord of length $4$ from a circle $C_{2}$ whose centre is $(3, -2)$. The radius of $C_{2}$ is
  1. $\sqrt {6}$
  2. $2$
  3. $\sqrt {2}$
  4. $3$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Tangent to C1 at (2,1): x(2)+y(1) - (x+2) - 1 = 0 -> x+y-3=0. Distance from (3,-2) to x+y-3=0 is |3-2-3|/sqrt(2) = 2/sqrt(2) = sqrt(2). Chord length 4 means half-chord is 2. Radius^2 = (sqrt(2))^2 + 2^2 = 2+4=6. Radius = sqrt(6).

Multiple choice

If a circle passing through the point $(-1, 0)$ touches y-axis at $(0, 2)$, then the length of the chord of the circle along the x-axis is
  1. $\displaystyle \frac{5}{2}$
  2. $5$
  3. $\displaystyle \frac{3}{2}$
  4. $3$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The circle touches the y-axis at (0, 2), so its center is (r, 2). Since it passes through (-1, 0), the distance from (r, 2) to (-1, 0) equals r. Solving (r+1)^2 + (2-0)^2 = r^2 gives r = -2.5. The circle equation is (x+2.5)^2 + (y-2)^2 = 6.25. Setting y=0, (x+2.5)^2 = 6.25 - 4 = 2.25, so x+2.5 = +/- 1.5. The chord length is |(-1) - (-4)| = 3.

Multiple choice

If $A\left( \cfrac { \pi }{ 3 } \right) ,B\left( \cfrac { \pi }{ 6 } \right) $ are the points on the circle represented in parametric form with centre $(0,0)$ and radius $12$, then the length of the chord $AB$ is
  1. $6\left( \sqrt { 6 } -\sqrt { 2 } \right) $
  2. $6\left( \sqrt { 6 } -\sqrt { 3 } \right) $
  3. $\sqrt { 2 } \left( \sqrt { 3 } -1 \right) $
  4. $6\left( \sqrt { 3 } -1 \right) $
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Points are (R cos theta1, R sin theta1) and (R cos theta2, R sin theta2). R=12, theta1=pi/3, theta2=pi/6. Chord length = sqrt((x2-x1)^2 + (y2-y1)^2) = 2R sin((theta1-theta2)/2) = 24 * sin(pi/12). sin(pi/12) = sin(15) = (sqrt(6)-sqrt(2))/4. Length = 24 * (sqrt(6)-sqrt(2))/4 = 6(sqrt(6)-sqrt(2)).

Multiple choice

Match the following I. The centroid of the triangle formed by $(2, 3, -1), (5, 6, 3), (2, -3, 1)$ is a) $(2, 2, 2)$ II. The circumcentre of the triangle formed by $(1, 2, 3), (2, 3, 1), (3, 1, 2)$ is b) $(3, 1, 4)$ III. The orthocenter of the triangle formed by $(2, 1, 5), (3, 2, 3), (4, 0, 4)$ is c) $(1, 1, 0)$ IV. The incentre of the triangle formed by $(0, 0, 0), (3, 0, 0), (0, 4, 0)$ is d) $(3, 2, 1)$ e) $(0, 0, 0)$
  1. $I - d, II - a, III - b, IV - c$
  2. $I - a, II - b, III - c, IV - d$
  3. $I - d, II - e, III - b, IV - c$
  4. $I - d, II - a, III - e, IV - c$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Centroid of (2,3,-1), (5,6,3), (2,-3,1) is ((2+5+2)/3, (3+6-3)/3, (-1+3+1)/3) = (3, 2, 1). This matches I-d. Checking other options confirms the mapping.

Multiple choice

A circle with centre at (2, 4) is such that the line $x+ y + 2 =0$ cuts a chord of length 6. Find the radius of the circle.
  1. $\sqrt{41}$
  2. $\sqrt{11}$
  3. $\sqrt{21}$
  4. $\sqrt{31}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The distance d from the center (2, 4) to the line x + y + 2 = 0 is |2 + 4 + 2| / sqrt(1^2 + 1^2) = 8 / sqrt(2) = 4 * sqrt(2). The radius r, distance d, and half-chord length a = 3 form a right triangle: r^2 = d^2 + a^2 = (4 * sqrt(2))^2 + 3^2 = 32 + 9 = 41. Thus r = sqrt(41).