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 length of the chord of a circle is $8$cm and perpendicular distance between centre and the chord is $3$cm. Then the radius of the circle is equal to?

  1. 4 cm

  2. 5 cm

  3. 6 cm

  4. 8 cm
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

A perpendicular from the center to a chord bisects the chord. This forms a right triangle with the radius as the hypotenuse, half the chord (4 cm) as one leg, and the distance to the chord (3 cm) as the other leg. By Pythagorean theorem, r^2 = 4^2 + 3^2 = 16 + 9 = 25, so r = 5.

Multiple choice

The length of chord of circle with radius 10cm drawn at a distance of 8cm

  1. 12cm

  2. 10cm

  3. 14cm

  4. 30cm
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

A perpendicular from the center to the chord bisects the chord. This forms a right triangle with the radius (10 cm) as the hypotenuse and the distance (8 cm) as one leg. The half-chord length is sqrt(10^2 - 8^2) = sqrt(100 - 64) = sqrt(36) = 6 cm. The full chord length is 2 * 6 = 12 cm.

Multiple choice

The Chord of contact of tangents from a point $P$ to a circle passes through $Q$. If $l_1$ and $l_2$ are the lengths of the tangents from $P$ and $Q$ to the circle, then $PQ$ is equal to
  1. $\displaystyle \frac{l_1+l_2} {2}$
  2. $\displaystyle \frac{l_1-l_2} {2}$
  3. $\displaystyle \sqrt{l^2_1+l^2_2}$
  4. $\displaystyle {2}\sqrt{l^2_1+l^2_2}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

For a point P outside a circle, the length of the tangent is sqrt(S1). The chord of contact from P is the line polar of P. Using properties of poles and polars, the relationship between the lengths of tangents and the distance between points is derived as PQ = sqrt(l1^2 + l2^2).

Multiple choice

$M$ is a fixed point in a plane and a point $P$ moves in the same plane such that $PM =10 cm$. State: (i) The name of the figure formed, (ii) The length of the radius of the circle, (iii) The length of the diameter of the circle. Can a chord of length of $16 cm$ be drawn in this figure?
  1. $ (i)circle,\quad (ii) 10cm,\quad (iii) 20cm,\quad (iii) yes$
  2. $ (i)circle,\quad (ii)20cm,\quad (iii) 20cm,\quad (iii) no$
  3. $ (i)circle,\quad (ii)10cm,\quad (iii)10cm,\quad (iii) yes$
  4. $ (i)circle,\quad (ii)20cm,\quad (ii) 10cm,\quad (iii)yes$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

A point at a fixed distance of 10 cm from M traces a circle with centre M and radius 10 cm. Its diameter is 20 cm, and a chord of length 16 cm can be drawn because it is shorter than the diameter.

Multiple choice

An isosceles $\Delta ABC$ is inscribed in a circle. If AB= AC =$12 \sqrt{5}$ cm and BC = 24 cm. The radius of the circle is
  1. $30 cm$
  2. $20cm$
  3. $15cm$
  4. $25cm$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

In an isosceles triangle with sides 12*sqrt(5), 12*sqrt(5), and base 24, the height h to the base is sqrt((12*sqrt(5))^2 - 12^2) = sqrt(720 - 144) = sqrt(576) = 24. The circumradius R = (abc) / (4 * Area). Area = 1/2 * base * height = 1/2 * 24 * 24 = 288. R = (12*sqrt(5) * 12*sqrt(5) * 24) / (4 * 288) = (144 * 5 * 24) / 1152 = 17280 / 1152 = 15.

Multiple choice

In a circle of radius $10$ cm, a chord is drawn $6$ cm from the centre. If a chord half the length of the original chord were drawn, its distance in centimeters from the centre would be
  1. $\sqrt{84}$
  2. $9$
  3. $8$
  4. $3\pi $
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The radius r is 10 and the distance d1 is 6. Using Pythagoras, half the chord length is sqrt(10^2 - 6^2) = 8, so the full chord is 16. A chord half this length is 8, meaning half its length is 4. The distance d2 from the center is sqrt(10^2 - 4^2) = sqrt(100 - 16) = sqrt(84).

Multiple choice

AB and CD are two parallel chords of a circle such that AB = 10 cm and CD = 24 cm, If the chords are on the opposite sides of the centre and the distance between them is 17 cm, the radius of the circle is:

  1. 14 cm

  2. 10 cm

  3. 13 cm

  4. 15 cm
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Draw a perpendicular from the center to the chords. The distances from the center are sqrt(R^2 - 5^2) and sqrt(R^2 - 12^2). Since they are on opposite sides, the sum of these distances is 17. Solving sqrt(R^2 - 25) + sqrt(R^2 - 144) = 17 yields R = 13.

Multiple choice

$AB$ is a chord of the circle with center $O$ and radius $r$, $OD\pm AB$ meeting $AB$ at $ D$. If $AB =8$ cm and $OD =3$ cm, then $r$ equals
  1. $7$ cm
  2. $11$ cm
  3. $5$ cm
  4. $\sqrt { 73 } $ cm
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

A perpendicular from the center to a chord bisects the chord, so AD = 8/2 = 4 cm. The radius, the 3 cm distance to the chord, and the 4 cm half-chord form a right triangle. Therefore r^2 = 3^2 + 4^2 = 25, so r = 5 cm, option C.