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 number of tangents, a circle can have is

  1. infinity
  2. $0$
  3. $1$
  4. $2$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

A tangent to a circle is a line that touches the circle at exactly one point. Since there are infinitely many points on the circumference of a circle, there are infinitely many tangents.

Multiple choice

The length of a tangent drawn from a point $10$ cm away from the centre of the circle of radius $5$ cm is
  1. $5$ cm
  2. $5\sqrt{3}$ cm
  3. $2\sqrt{3}$ cm
  4. $\sqrt{15}$ cm
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The radius, tangent, and the line from the center to the external point form a right triangle. The distance to the center is the hypotenuse (10), and the radius is one leg (5). Tangent length = sqrt(10^2 - 5^2) = sqrt(100 - 25) = sqrt(75) = 5*sqrt(3).

Multiple choice

In figure there are two concentric circles with centre O and of radii 5 cm and 3 cm. From an external point P tangents PA and PB are drawn to these circles. If AP = 12 cm find the length of BP
  1. $\displaystyle 4\sqrt{10}$ cm
  2. $\displaystyle 2\sqrt{10}$ cm
  3. $\displaystyle \sqrt{10}$ cm
  4. $\displaystyle 3\sqrt{10}$ cm
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

In the outer circle, radius R=5, tangent AP=12. By Pythagoras, OP^2 = 5^2 + 12^2 = 25 + 144 = 169, so OP = 13. In the inner circle, radius r=3, tangent BP. By Pythagoras, OP^2 = r^2 + BP^2. 169 = 3^2 + BP^2. 169 = 9 + BP^2. BP^2 = 160. BP = sqrt(160) = 4 * sqrt(10).

Multiple choice

The radius of the circle which has the lines ${x}+{y}-1=0$ and ${x}+{y}-9=0$ as tangents is
  1. $\sqrt{2}$
  2. $2\sqrt{2}$
  3. $3\sqrt{2}$
  4. $4\sqrt{2}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The distance between two parallel lines Ax + By + C1 = 0 and Ax + By + C2 = 0 is given by |C1 - C2| / sqrt(A^2 + B^2). Here, the lines are x + y - 1 = 0 and x + y - 9 = 0. The distance is | -1 - (-9) | / sqrt(1^2 + 1^2) = 8 / sqrt(2) = 4 * sqrt(2). The diameter of the circle is this distance, so the radius is half of that, which is 2 * sqrt(2).

Multiple choice

A point P is $26 cm$ away from the centre O of a circle and the length PT of tangent drawn from P to the circle is $10 cm$. Then the radius of the circle is
  1. $18 cm$
  2. $20 cm$
  3. $22 cm$
  4. $24 cm$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

A tangent to a circle is perpendicular to the radius at the point of contact. This forms a right-angled triangle with the hypotenuse as the distance from the center to the external point (26 cm) and one leg as the tangent (10 cm). Using the Pythagorean theorem, r^2 + 10^2 = 26^2, so r^2 = 676 - 100 = 576, giving r = 24 cm.

Multiple choice

Two tangents are drawn to a circle from an external point $A$, touching the circle at the points $P$ and $Q$. A third tangent intersects segment $AP$ at $B$ and segment $AQ$ at $C$ and touches the circle at $R$. If $AQ = 10$ units, then the perimeter of $\Delta ABC$ is
  1. $22.0$
  2. $20.5$
  3. $20.0$
  4. $40.0$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

By the property of tangents from an external point, the lengths of tangents from A to the circle are equal (AP = AQ = 10). The perimeter of triangle ABC is AB + BC + AC. Since the third tangent touches at R, BR = BP and CR = CQ. Thus, the perimeter is AB + (BR + RC) + AC = (AB + BP) + (AC + CQ) = AP + AQ = 10 + 10 = 20.

Multiple choice

In fig 9.4, AB is a chord of the circle and AOC is its diameter such that ${\angle ACB= 50 ^0}$. If AT is the tangent to the circle at the point A, then ${\angle BAT}$ is equal to
  1. ${65 ^0}$
  2. ${60^ 0}$
  3. ${50 ^0}$
  4. ${40^0}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Angle in a semicircle is 90 degrees, so angle ABC = 90. In triangle ABC, angle BAC = 180 - 90 - 50 = 40. Angle between tangent AT and chord AB is equal to the angle in the alternate segment, which is angle ACB = 50.

Multiple choice

In fig 9.6, if O is the centre of a circle, PQ is chord and the tangent PR at P makes an angle of ${50^0}$ with PQ, then ${\angle POQ}$ is equal to
  1. ${100^0}$
  2. ${80^0}$,
  3. ${90^0}$
  4. ${75^0}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The angle between a tangent and a chord is equal to the angle in the alternate segment. If angle between tangent PR and chord PQ is 50, the angle in the alternate segment is 50. The central angle subtended by the same chord is twice the angle in the alternate segment, so 2 * 50 = 100 degrees.