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

To draw a pair of tangents to a circle which are inclined to each other at an angle $\displaystyle { 35 }^{ \circ }$, it is required to draw tangents at the end points of those two radii of the circle, the angle between which is
  1. $\displaystyle { 105 }^{ \circ }$
  2. $\displaystyle { 70 }^{ \circ }$
  3. $\displaystyle { 145 }^{ \circ }$
  4. $\displaystyle { 150 }^{ \circ }$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The angle between two tangents to a circle and the angle between the radii at the points of contact are supplementary. If the tangents are inclined at 35 degrees, the angle between the radii is 180 - 35 = 145 degrees.

Multiple choice

Draw a circle with centre $C$ and radius $3\ cm$. Take a point $P$ outside the circle such that $CP = 6\ cm$. Construct tangents $PA$ and $PB$ from this point to the circle, where $A$ and $B$ are the intersection points of the tangents. Then $m \angle ACB = $ ?
  1. $60^\circ$
  2. $120^\circ$
  3. $30^\circ$
  4. $90^\circ$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

In right triangle OAP (where O is the center), OA = 3 and OP = 6. Thus, sin(angle OPA) = 3/6 = 1/2, so angle OPA = 30 degrees. In triangle OAP, angle AOP = 60 degrees. Since the two tangents are symmetric, angle AOB = 2 * 60 = 120 degrees.

Multiple choice

Draw a circle of radius $3\ cm$. Take a point outside the circle at a distance of $6\ cm$ from the centre of the circle. Construct tangents from this point to the circle. Measure the angle between the tangents and select the correct value from below.
  1. $60^\circ$
  2. $30^\circ$
  3. $45^\circ$
  4. $90^\circ$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

In the right-angled triangle formed by the center of the circle, the point of tangency, and the external point, the sine of half the angle between the tangents is equal to the radius divided by the distance to the center, which is 3/6 = 1/2. This means half the angle is 30 degrees, so the total angle between the tangents is 60 degrees.

Multiple choice

Draw a circle with center $O$ and radius $6$ cm. Take a point $P$ outside the circle at a distance of $10$ cm from $O$. Draw tangents to the circle from point $P$. Let the tangents intersect the circle in points $A$ and $B$. Find the approximate value of $\angle OPB$.
  1. $37^o$
  2. $53^o$
  3. $45^o$

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

In the right triangle formed by the center O, the point P, and the tangent point A, OA = 6, OP = 10. Thus, AP = sqrt(10^2 - 6^2) = 8. In triangle OAP, sin(angle OPA) = OA/OP = 6/10 = 0.6. Angle OPA = arcsin(0.6) approx 36.87 degrees. Since the two tangents are symmetric, angle OPB is the same.

Multiple choice

Draw a circle with centre $O$ and radius $6$ cm. Take a point $P$ outside the circle at a distance of $10$ cm from $O$. Draw tangents to the circle from point $P$. Let the tangents intersect the circle in points $A$ and $B$. Find the area of triangle $OBP$ in sq. cm.
  1. $24$
  2. $26$
  3. $25$

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

In the right triangle OAP (where A is the point of tangency), OA = 6, OP = 10. By Pythagoras, AP = sqrt(10^2 - 6^2) = 8. The area of triangle OAP is 1/2 * 6 * 8 = 24. Since triangle OBP is congruent to OAP, its area is also 24.

Multiple choice

Draw a circle with centre $O$ and radius $6$ cm. Take a point $P$ outside the circle at a distance of $10$ cm from $O$. Draw tangents to the circle from point $P$. Let the tangents intersect the circle in points $A$ and $B$. Find $BP$.
  1. $6$ cm
  2. $8$ cm
  3. $8.5$ cm

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

Triangle OAP is a right triangle with hypotenuse OP=10 and leg OA=6. By Pythagoras, AP = sqrt(10^2 - 6^2) = 8. Since tangents from a point to a circle are equal, BP = AP = 8.

Multiple choice

Consider the following statements : 1. If $3 \theta $ is an acute angle such that $\sin 3 \theta = \cos 2 \theta$, then the measurement of $\theta$ in radian equals to $\frac{\pi}{10}$. 2. One radian is the angle subtended at the centre of a circle by an arc of the same circle whose length is equal to the diameter of that circle. Which of the above statements is/are correct ?

  1. 1 only

  2. 2 only

  3. Both 1 and 2

  4. Neither 1 nor 2
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Since 3θ is acute, sin 3θ = cos 2θ gives 3θ = 90° - 2θ, so 5θ = 90° and θ = π/10. One radian corresponds to an arc whose length equals the radius, not the diameter, so statement 2 is false.

Multiple choice

Find the difference of the areas of two segments of a circle formed by a chord of length $5\ cm$ . Subtending an angle of $90^{o}$ till the centre.
  1. $32.14$ sq. cm
  2. $35.42$ sq. cm
  3. $38.96$ sq. cm
  4. $42.43$ sq. cm
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Area of segment = Area of sector - Area of triangle. Sector angle = 90 degrees. Area of sector = (90/360) * pi * r^2. Chord length = 5, so r^2 + r^2 = 5^2 => 2r^2 = 25 => r^2 = 12.5. Area of sector = 0.25 * 3.14 * 12.5 = 9.8125. Area of triangle = 0.5 * r^2 = 6.25. Area of segment = 9.8125 - 6.25 = 3.5625. The difference between the two segments is (pi * r^2 - 3.5625) - 3.5625 = 39.25 - 7.125 = 32.125.

Multiple choice

In figure $A,B$ and $C$ are three points on a circle with centre $O$ such that $\angle{BOC}={30}^{o}$ and $\angle {AOB}={60}^{o}$. If $D$ is a point on the circle other than the arc $ABC$, find $\angle {ADC}$.
  1. $30^\circ$
  2. $90^\circ$
  3. $45^\circ$
  4. $60^\circ$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The central angle subtended by arc AC is angle AOC = angle AOB + angle BOC = 60 + 30 = 90 degrees. The angle subtended by an arc at the center is double the angle subtended at any point on the remaining part of the circle. Therefore, angle ADC = 1/2 * angle AOC = 1/2 * 90 = 45 degrees.