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

In a circle of radius $13$cm, $PQ$ and $RS$ are two parallel chords of length $24$cm and l0cm respectively. The chords are on the opposite sides of the centre. The distance between the chords is?
  1. $7$cm
  2. $17$cm
  3. $26$cm
  4. $12$cm
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The distance from the centre to a chord is sqrt(r^2 - (chord/2)^2). For the 24 cm chord this is 5 cm, and for the 10 cm chord it is 12 cm. Since the chords lie on opposite sides, their separation is 5 + 12 = 17 cm.

Multiple choice

The radius of a circle with centre $O$ is $41$. The length of the chord $PQ$ of same circle is $80$. The distance of chord $PQ$ from centre $O$ is
  1. $81$ cm
  2. $27$ cm
  3. $3$ cm
  4. $9$ cm
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

In a circle, the perpendicular from the center to a chord bisects the chord. This forms a right triangle with hypotenuse = radius (41), base = half-chord (40), and height = distance (d). d^2 + 40^2 = 41^2. d^2 + 1600 = 1681. d^2 = 81, d = 9.

Multiple choice

A chord of length $30$ cm is drawn at a distance of $8$ cm from the centre of a circle. The radius of the circle (in cm.) is
  1. $15$ cm
  2. $21$ cm
  3. $18$ cm
  4. $17$ cm
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

A perpendicular from the center to a chord bisects the chord. This forms a right-angled triangle with the distance from the center (8 cm) as one leg, half the chord (15 cm) as the other leg, and the radius as the hypotenuse. Using the Pythagorean theorem, r^2 = 8^2 + 15^2 = 64 + 225 = 289, so r = 17 cm.

Multiple choice

A chord $AB$ is at a distance of $6$ cm from the centre of a circle whose radius is $6$ cm less than that of the chord $AB$. Then the length of the chord $AB$ is
  1. $8$ cm
  2. $32$ cm
  3. $24$ cm
  4. $16$ cm
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Let r be the radius of the circle. The distance from the center to the chord is 6. The chord length is L. The half-chord length is L/2. By Pythagoras, r^2 = 6^2 + (L/2)^2. We are given r = L - 6. Substituting gives (L-6)^2 = 36 + L^2/4. Solving for L gives L = 16.

Multiple choice

The radius of a circle is $16cm$. The mid point of a chord of the circle lies on the diameter perpendicular to the chord and its distance from the near end of the diameter is $3cm$; If the length of that chord is $m \sqrt{87}$ cm, then the value of $m$ is
  1. $4$
  2. $8$
  3. $2$
  4. $1$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let the diameter be AB. The chord is perpendicular to the diameter at a point P. Distance from near end is 3, so distance from center is 16-3 = 13. In the right triangle formed by radius (16), distance to center (13), and half-chord (h), h^2 = 16^2 - 13^2 = 256 - 169 = 87. Half-chord = sqrt(87). Full chord = 2*sqrt(87). So m=2.

Multiple choice

A chord of the circle $x^{2} +y^{2}-4x -6y = 0$ passing through the origin subtends an angle $\tan^{-1} \left(\dfrac{7}{4}\right)$ at the point where the circle meets positive y-axis. Equation of the chord is
  1. $ 2x + 3y = 0$
  2. $x + 2y = 0$
  3. $ x -2y = 0 $
  4. $ 2x -3y = 0$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The circle is (x-2)^2 + (y-3)^2 = 13. The center is (2,3). The circle meets the positive y-axis at (0,6). The chord passes through (0,0) and (0,6) is not possible as it is a chord. The line x-2y=0 passes through the origin and satisfies the geometric conditions.