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 $O$ is the circumcentre of the $\Delta ABC$ and $R_1,R_2$ and $R_3$ are the radii of the circumcircles $OBC,\,\,OCA$ and $OAB$ repectively then $\dfrac{a}{{{R_1}}} + \dfrac{b}{{{R_2}}} + \dfrac{c}{{{R_3}}}$ has the value equal to:
  1. $\dfrac{{abc}}{{2{R^3}}}$
  2. $\dfrac{{{abc}}}{{R^3}}$
  3. $\dfrac{{4\Delta }}{{{R^2}}}$
  4. $\dfrac{\Delta }{{4{R^2}}}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

In any triangle, the ratio a/R1 = 2*sin(A), b/R2 = 2*sin(B), c/R3 = 2*sin(C). Using the properties of circumcenters and circumradii, the sum simplifies to abc / R^3.

Multiple choice

Area of the cyclic square is 256 cm $^{2}$, then what is the area of the circle inscribing the square
  1. 64$\pi cm^{2}$
  2. 118$\pi cm^{2}$
  3. 136$\pi cm^{2}$
  4. 128$\pi cm^{2}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Area of square = s^2 = 256, so side s = 16. The diagonal of the square is the diameter of the circumscribing circle. Diagonal d = s*sqrt(2) = 16*sqrt(2). Radius r = d/2 = 8*sqrt(2). Area of circle = pi * r^2 = pi * (8*sqrt(2))^2 = pi * 64 * 2 = 128*pi.

Multiple choice

In a circle of radius 21 cm, an arc subtends an angle of $60^{\circ}$ at the centre. The area of the segment formed by the corresponding chord of the arc is :
  1. 42 $cm^2$
  2. 421.73 $cm^2$
  3. 429.43 $cm^2$
  4. 40 $cm^2$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Area of segment = Area of sector - Area of triangle. Sector area = (60/360) * pi * 21^2 = (1/6) * (22/7) * 441 = 231. Triangle area = (sqrt(3)/4) * 21^2 = (1.732/4) * 441 = 190.95. Segment area = 231 - 190.95 = 40.05. Option D is the closest.

Multiple choice

The area of a circle is $\displaystyle 220 \text{cm}^{2}$. The side of the square inscribed in this circle will be ?
  1. $\sqrt{140} \text{cm}$
  2. $\sqrt{210} \text{cm}$
  3. $\sqrt{70} \text{cm}$
  4. $\sqrt{35} \text{cm}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Area of circle = pi * r^2 = 220. r^2 = 220/pi. For a square inscribed in a circle, the diagonal of the square is the diameter of the circle. d = 2r. Diagonal = s * sqrt(2) = 2r. s^2 * 2 = 4r^2. s^2 = 2r^2 = 2 * (220/pi) = 440/pi. Using pi = 22/7, s^2 = 440 / (22/7) = 140. So s = sqrt(140).

Multiple choice

The area of a segment of a circle of radius $21 cm$, if the arc of the segment has a measure of $\displaystyle 60^{\circ}$ is: (Take $\sqrt{3}=1.73 $)
  1. $\displaystyle 45.27\:cm^{2}$
  2. $\displaystyle 40.27\:cm^{2}$
  3. $\displaystyle 40.8\:cm^{2}$

  4. None of the above
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Area of segment = Area of sector - Area of triangle. Sector area = (60/360) * pi * 21^2 = (1/6) * (22/7) * 441 = 231. Triangle area = (sqrt(3)/4) * 21^2 = 0.4325 * 441 = 190.73. Segment area = 231 - 190.73 = 40.27.