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 chord of least length which passes through the point $(2,1)$ of the circle $\mathrm{x}^{2}+\mathrm{y}^{2}- 2\mathrm{x}-4\mathrm{y}-13=0$ is
  1. $2$
  2. $3\sqrt{2}$
  3. $2\sqrt{3}$
  4. $8$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The chord of least length passing through a point inside a circle is the one perpendicular to the diameter passing through that point. However, the question asks for the length of the chord passing through (2, 1). The center of the circle x^2 + y^2 - 2x - 4y - 13 = 0 is (1, 2) and the radius is sqrt(1^2 + 2^2 + 13) = sqrt(18) = 3*sqrt(2). The distance from (1, 2) to (2, 1) is sqrt((2-1)^2 + (1-2)^2) = sqrt(2). The half-length of the chord is sqrt(r^2 - d^2) = sqrt(18 - 2) = 4. The full length is 8.

Multiple choice

I) lf a square of side $10$ is inscribed in a circle, then the radius of the circle is $5$ $\sqrt{2}$ II) lf a circle is inscribed in a square of side $10$ so that the circle touches the $4 $ sides internally, then radius is $5$. Which of above statements is true?

  1. Only I

  2. Only II

  3. both I and II

  4. neither I nor II
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

I: Square side 10, diagonal = 10*sqrt(2). If inscribed in a circle, diagonal is diameter, so radius = 5*sqrt(2). Correct. II: Circle inscribed in square side 10, diameter = 10, radius = 5. Correct.

Multiple choice

$AB$ and $CD$ are two parallel chords of circle such that $AB = 10$ cm and $CD = 24$ cm. If $LM = 17$ cm, then the diameter of the circle is
  1. $13$ cm
  2. $26$ cm
  3. $14$ cm

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

Let r be the radius. The distances from the center to the chords are d1 = sqrt(r^2 - 5^2) and d2 = sqrt(r^2 - 12^2). Since chords are on the same side, d1 - d2 = 17 (or d1 + d2 = 17). Solving sqrt(r^2 - 25) - sqrt(r^2 - 144) = 17 leads to r = 13. Diameter = 2r = 26.

Multiple choice

A quadrilateral ABCD is inscribed in a circle so that AB is the diameter of the circle. If $\angle$ ADC = 115$^{\circ}$, find $\angle$ BAC.
  1. 20$^{\circ}$
  2. 25$^{\circ}$
  3. 30$^{\circ}$
  4. 40$^{\circ}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

In a cyclic quadrilateral, opposite angles sum to 180 degrees. If angle ADC = 115, then angle ABC = 180 - 115 = 65 degrees. Since AB is the diameter, angle ACB = 90 degrees (angle in a semicircle). In triangle ABC, angle BAC = 180 - 90 - 65 = 25 degrees.

Multiple choice

A chord of the circle $x^{2} + y^{2} = 8$ makes equal intercepts on the coordinate axes. If the length of each intercept is '$a$' then the range of '$a$' is
  1. $(-2, 2)$
  2. $(-4, 4)$
  3. $(-2\sqrt{2},2\sqrt{2})$
  4. $( -1, 1)$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The chord makes equal intercepts 'a' on axes, so it lies on x + y = a or x - y = a. Distance from origin to x + y - a = 0 must be less than radius sqrt(8). Distance = |a| / sqrt(2) < sqrt(8) = 2 * sqrt(2). So |a| < 4, meaning -4 < a < 4.

Multiple choice

If the length of the common chord of two intersecting equal circles be 6 cm and if the radius of each circle be 5 cm then the distance between the centers of the circle is

  1. 7 cm

  2. 8 cm

  3. 9 cm

  4. none
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The common chord divides the line connecting the centers into two equal parts (3 cm each) and is perpendicular to it. Using the Pythagorean theorem in one of the triangles formed by the radius, half the chord, and the distance from the center to the chord: d^2 + 3^2 = 5^2, so d = 4. The distance between centers is 2d = 8 cm.

Multiple choice

The distance of a chord 8 cm from the centre of a circle of radius 5 cm is
  1. $4\ cm$
  2. $3\ cm$
  3. $2\ cm$
  4. $9\ cm$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

A chord of length 8 cm is bisected by the perpendicular from the center, creating a right triangle with the radius (5 cm) as the hypotenuse and half the chord (4 cm) as one leg. The distance from the center is the other leg, calculated as sqrt(5^2 - 4^2) = sqrt(25 - 16) = sqrt(9) = 3 cm.

Multiple choice

$AB$ is a chord of a circle. Tangent $MC$ touches the circle at $M$ and meets $AB$ produced at $C$. If $MC = 12 cm, AB = xcm$ and $BC= (x-2) cm$ then the value of $x$ is equal to ?
  1. $10 cm$
  2. $12 cm$
  3. $13 cm$
  4. $14 cm$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

By the tangent-secant theorem, MC^2 = BC * AC. Given MC = 12, BC = x-2, AC = AB + BC = x + (x-2) = 2x-2. So 144 = (x-2)(2x-2) => 144 = 2(x-2)(x-1) => 72 = x^2 - 3x + 2 => x^2 - 3x - 70 = 0. Factoring gives (x-10)(x+7) = 0. Since x must be positive, x = 10.

Multiple choice

A circle has two equal chords $PQ $ and $ PR $ , diameter $ PD $ cuts $ QR $ in $ E $ . If $PR = 12 cm$ and $PE = 8 cm$, then the length of $ PD $ is ?
  1. $25$ cm
  2. $22$ cm
  3. $20$ cm
  4. $18$ cm
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Let O be the center. PQ=PR, so O lies on the angle bisector of QPR. PD is diameter, so PD passes through O. Triangle PQR is isosceles. E is on QR. Since PD is diameter and perpendicular to chord QR (as it bisects it), PE is the altitude. In right triangle PQR, if we use properties of chords, the length of PD can be determined by geometric relations.