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 PQ is a chord of a circle whose centre is 0 and PR is the tangent to the circle at the point P, then $\angle POQ$ is equal to
  1. $\angle RPQ$
  2. $2\angle RPQ$
  3. $3\angle RPQ$
  4. $4\angle RPQ$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let angle OPQ = theta. Since PR is tangent, angle OPR = 90. Angle RPQ = 90 - theta. In triangle OPQ, OP = OQ (radii), so angle OQP = angle OPQ = theta. Angle POQ = 180 - 2 * theta. This doesn't directly relate to RPQ unless we use the property that the angle between tangent and chord equals the angle in the alternate segment. Angle RPQ = angle in alternate segment. This is a standard circle theorem.

Multiple choice

Two congruent circles of radius r intersect such that each passes through the centre of the other, then the length of the common chord is given by ?
  1. $r$
  2. $2r$
  3. $\sqrt 2r$
  4. $\sqrt 3r$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Two circles of radius r intersect such that each passes through the center of the other. The distance between centers is r. The common chord forms two equilateral triangles with the centers, each with side r. The height of one triangle is (sqrt(3)/2) * r. The total length of the chord is 2 * (sqrt(3)/2) * r = sqrt(3) * r.

Multiple choice

Radii of two concentric circles are 40 cm and 41 cm. AB is a chord of the bigger circle and tangent to the smaller circle. Find the length of AB
  1. $9$
  2. $18$
  3. $13$
  4. $26$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The radius of the smaller circle (40) is perpendicular to the chord AB at the point of tangency, bisecting it. This forms a right triangle with hypotenuse 41 and one leg 40. The other leg (half of AB) = sqrt(41^2 - 40^2) = sqrt(1681 - 1600) = sqrt(81) = 9. AB = 2 * 9 = 18.

Multiple choice

Two circles with centres O and O' of radii 3 cm and 4 cm, respectively intersect at two points P and Q such that OP and O'P are tangents to the two circles. Find the length of the common chord PQ.

  1. 4.8

  2. 5.4

  3. 3.2

  4. 7.9
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let O=(0,0), O'=(d,0). Radii 3, 4. Tangents at P imply triangle OPO' is right-angled at P. O'P^2 + OP^2 = OO'^2. 3^2 + 4^2 = 5^2. Distance OO' = 5. Height of triangle OPO' to hypotenuse = (3*4)/5 = 2.4. Common chord PQ = 2 * 2.4 = 4.8.

Multiple choice

In a circle whose radius is $10$ inches, a chord is $6$ inches from the center. What is the length of the chord?
  1. $4$ inches
  2. $6$ inches
  3. $8$ inches
  4. $16$ inches
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The perpendicular from the center of a circle to a chord bisects the chord, forming a right triangle where the hypotenuse is the radius (10 inches) and one leg is the distance from the center (6 inches). Using the Pythagorean theorem, the other leg (half the chord length) is sqrt(10^2 - 6^2) = 8 inches, making the total chord length 16 inches.

Multiple choice

Two parallel chords of a circle with diameter $13\ cm$ are, $5\ cm$ and $12\ cm$ in length. If both the chords lies in a semi-circle, find the distance between the chords
  1. $8.5\ cm$
  2. $5\ cm$
  3. $3.5\ cm$
  4. $3\ cm$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The radius of the circle is 6.5 cm. The distance of the 5 cm chord from the center is sqrt(6.5^2 - 2.5^2) = 6 cm. The distance of the 12 cm chord from the center is sqrt(6.5^2 - 6^2) = 2.5 cm. Since they are on the same side of the center, the distance between them is 6 - 2.5 = 3.5 cm.

Multiple choice

The locus of the midpoints of chords of the circle $x^2+y^2 = 1$ which subtends a right angle at the origin is
  1. $x^2+y^2 = \dfrac{1}{4}$
  2. $x^2+y^2 = \dfrac{1}{2}$
  3. $xy = 0$
  4. $x^2-y^2 = 0$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

For a chord to subtend a right angle at the origin, the distance from the origin to the chord must be 1/sqrt(2). The locus of the midpoint (h, k) of such chords is x^2 + y^2 = d^2, where d is the distance from the origin. Thus, x^2 + y^2 = (1/sqrt(2))^2 = 1/2.

Multiple choice

The centre of the circle inscribed in a square formed by the lines $x^2 - 8x + 12 = 0$ and $ y^2 - 14y + 45 = 0$ is
  1. $(4, 7)$
  2. $(7, 4)$
  3. $(9, 4)$
  4. $(4, 9)$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The lines x^2 - 8x + 12 = 0 factor to (x-2)(x-6)=0, so x=2 and x=6. The lines y^2 - 14y + 45 = 0 factor to (y-5)(y-9)=0, so y=5 and y=9. This forms a square with vertices (2,5), (6,5), (6,9), (2,9). The center is the midpoint of the diagonals, which is ((2+6)/2, (5+9)/2) = (4, 7).

Multiple choice

The length of the chord cut off by $y = 2x + 1$ from the circle $x^2 + y^2 = 2$ is
  1. $3/8$
  2. $8/7$
  3. $6/\sqrt{5}$
  4. $5/6$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The circle is x^2 + y^2 = 2, so radius r = sqrt(2). The distance d from the center (0,0) to the line 2x - y + 1 = 0 is |2(0) - 0 + 1| / sqrt(2^2 + (-1)^2) = 1 / sqrt(5). The half-length of the chord is sqrt(r^2 - d^2) = sqrt(2 - 1/5) = sqrt(9/5) = 3 / sqrt(5). The total length is 2 * (3 / sqrt(5)) = 6 / sqrt(5).

Multiple choice

Let AB be a chord of the circle $x^2 + y^2 = r^2$ subtending a right angle at the centre. Then the centroid of the triangle PAB as P moves on the circle is

  1. a parabola

  2. an ellipse

  3. a circle

  4. a pair of straight lines
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let A = (r, 0) and B = (0, r) on the circle x^2 + y^2 = r^2. Let P = (r cos t, r sin t). The centroid G = ((r + 0 + r cos t)/3, (0 + r + r sin t)/3). This simplifies to (x - r/3)^2 + (y - r/3)^2 = (r/3)^2, which is the equation of a circle.

Multiple choice

The locus of the midpoints of the chords of the circle $\left ({ x }^{ 2 }+{ y }^{ 2 }-ax-by\right )=0$ which subtend a right angle at $\left (\dfrac {a}{2},\dfrac {b}{2}\right )$ is
  1. $ax+by=0$
  2. $ax+by={ a }^{ 2 }+{ b }^{ 2 }$
  3. ${ x }^{ 2 }+{ y }^{ 2 }-ax-by+\cfrac { { a }^{ 2 }+{ b }^{ 2 } }{ 8 } =0$
  4. ${ x }^{ 2 }+{ y }^{ 2 }-ax-by-\cfrac { { a }^{ 2 }-{ b }^{ 2 } }{ 8 } =0$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The locus of midpoints of chords subtending a right angle at a point is derived from the circle geometry properties. The correct equation is x^2 + y^2 - ax - by + (a^2 + b^2)/8 = 0.