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

Let x be the area of a square inscribed in a circle of radius r and y be the area of an equilateral triangle inscribed in the same circle. Which one of the following is correct?

  1. 9x2 = 16y2

  2. 27x2 = 64y2

  3. 36x2 = 49y2

  4. 16x2 = 21y2
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Square in circle of radius r: diagonal = 2r, side = sqrt(2)r, area x = 2r^2. Equilateral triangle in circle of radius r: side = sqrt(3)r, area y = (sqrt(3)/4) * (sqrt(3)r)^2 = (3*sqrt(3)/4)r^2. x^2 = 4r^4. y^2 = (27/16)r^4. x^2/y^2 = 4 / (27/16) = 64/27. 27x^2 = 64y^2.

Multiple choice

Question: What is the area of the circle C? Statement I: An arc of length 7 cm subtends an angle 30° at the centre of C. Statement II: A chord of length 10 cm subtends an angle 90° at the centre of C.

  1. The Question can be answered by one of the Statements alone but not by the other.

  2. The Question can be answered by either Statement alone.

  3. The Question can be answered by using both the Statements together, but cannot be answered by using either Statement alone.

  4. The Question cannot be answered even by using both Statements together.
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The area of a circle is pi * r^2. Statement I gives arc length L = r * theta (in radians). With L = 7 and theta = 30 degrees (pi/6 radians), r = 7 / (pi/6) = 42/pi. This allows calculating the area. Statement II gives a chord length c = 2 * r * sin(theta/2). With c = 10 and theta = 90 degrees, 10 = 2 * r * sin(45), so 10 = 2 * r * (sqrt(2)/2), which means r = 10/sqrt(2). This also allows calculating the area. Since both statements independently provide enough information to find the radius, the answer is B.

Multiple choice

In the right-angled triangle XYZ, in which the hypotenuse XZ has a length of 30 cm and a perpendicular line segment YN is constructed on side XZ, what is the maximum length in centimetres for the line segment YN?

  1. 5

  2. 10

  3. 15

  4. 20
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

In a right triangle, the altitude to the hypotenuse is maximized when the triangle is isosceles. In an isosceles right triangle with hypotenuse 30, the altitude to the hypotenuse is half the hypotenuse, which is 15.

Multiple choice

Given two concentric circles with centre P, AC is a chord of the larger circle tangent to the smaller at B. If |BC| = 2 units, find the area of the annulus between the circles in square units.

  1. 4π

  2. 6π

  3. 8π

  4. 16π
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let R, r be the radii of bigger & smaller circles respectively. In ΔABP R2 = r2 + (AB)2 ………… (1) As BC = 2 Ac is a chord PB is perpendicular to it & must bisect the chord. AB = BC = 2 From (1) R2 - r2 = 4 Area of the annulus = πR2 - πr2 = π(R2 - r2) = 4π

Multiple choice

In a circle with radius 8 centimeters, there are two parallel chords. The first chord is positioned 3 centimeters away from the center of the circle, and the second chord is positioned 5 centimeters away from the center. Quantity A: Distance between the chords Quantity B: 6.93 centimeters

  1. Quantity A is greater.

  2. Quantity B is greater.

  3. The two quantities are equal.

  4. The relationship cannot be determined from the information given.
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Distance from center to chord 1 is d1=3, chord 2 is d2=5. Half-length of chord 1 is sqrt(8^2 - 3^2) = sqrt(55) approx 7.41. Half-length of chord 2 is sqrt(8^2 - 5^2) = sqrt(39) approx 6.24. The distance between chords depends on whether they are on the same or opposite sides of the center. If same side, distance = 5-3 = 2. If opposite, distance = 5+3 = 8. Neither is 6.93.

Multiple choice

Two parallel chords AB and CD of a circle are of lengths 60 cm and 80 cm respectively. They are on the same side of the centre O and 10 cm apart. What is the diameter of the circle?

  1. 120 cm

  2. 110 cm

  3. 100 cm

  4. 90 cm
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Using the properties of chords and the Pythagorean theorem, let r be the radius. The distances from the center to the chords are d1 and d2. We have r^2 = 30^2 + d1^2 and r^2 = 40^2 + d2^2. Given d2 - d1 = 10, solving the system yields r = 50. The diameter is 2r = 100 cm.