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

Consider a circle with unit radius. There are 7 adjacent sectors, S1, S2, S3, ..., S7 in the circle such that their total area is (1/8)th of the area of the circle. Further, the area of the jth sector is twice the area of the (j - 1)th sector, for j = 2, …, 7. What is the measure of the angle (in radians) subtended by the arc of S1 at the centre of the circle?

  1. π/508

  2. π/2040

  3. π/1016

  4. π/1524

  5. None of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Total area = (1/8) * pi * r^2 = pi/8 (since r=1). Let the area of S1 be A. Then A + 2A + 4A + ... + 64A = pi/8. A(2^7 - 1)/(2-1) = pi/8. 127A = pi/8, so A = pi/1016. The area of a sector is (theta/2) * r^2. Since r=1, A = theta/2. So theta/2 = pi/1016, theta = pi/508.

Multiple choice

A tangent to a circle of radius 3 cm from an external point P is of length 4 cm. Find the distance of P from the nearest point of the circumference.

  1. 2 cm

  2. 3 cm

  3. 4 cm

  4. 5 cm

  5. 6 cm
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The distance from the center to the external point P is found using the Pythagorean theorem: sqrt(3^2 + 4^2) = 5 cm. The distance from P to the nearest point on the circumference is the distance from P to the center minus the radius, which is 5 - 3 = 2 cm.

Multiple choice

A, B and C are three consecutive points on the arc of a semi-circle, such that the angles subtended by the chords AB and AC at the centre of the circle are 60 and 100 degrees, respectively. Find the measure of angle BAC.

  1. 30o

  2. 50o

  3. 25o

  4. 20o

  5. 15o
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The angles subtended by chords AB and AC at the center are 60 and 100 degrees. The angle subtended by chord BC at the center is 100 - 60 = 40 degrees. The angle subtended by a chord at the circumference is half the angle subtended at the center. Angle BAC is the angle subtended by chord BC at the circumference, which is 40 / 2 = 20 degrees.

Multiple choice

The quadrilateral ABCD is such that it circumscribes a circle. The circle inscribed in the triangle ABC touches AC at X and the circleinscribed in the triangle ADC touches AC at Y. Find the value of XY/AC.

  1. 0

  2. 1

  3. 2

  4. 3
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The total age of 10 students is 10 * 25 = 250 years. After a 16-year-old student leaves, the total age becomes 250 - 16 = 234 years, and the number of students becomes 9. The new average is 234 / 9 = 26 years.

Multiple choice

Directions: Compare the quantities and mark your answers as: In a circle of radius 5 cm, there are two parallel chords of lengths 6 cm and 4 cm, respectively. Quantity A Quantity B The distance between the chords 8.58 cm

  1. If quantity A is greater

  2. If quantity B is greater

  3. If both quantities are equal

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

The distance between parallel chords depends on whether they are on the same side or opposite sides of the center. With radius 5, the distances from the center are sqrt(5^2 - 3^2) = 4 and sqrt(5^2 - 2^2) = sqrt(21) approx 4.58. The distance between chords is either 4.58 - 4 = 0.58 or 4.58 + 4 = 8.58. Since both are possible, the relationship cannot be determined.

Multiple choice

What is the length of the hypotenuse of right triangle ABC? (A) Perimeter of ABC is 60 cm. (B) Rectangle contained by the two sides containing the right angle is 300 cm2.

  1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

  2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

  3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

  4. EACH statement ALONE is sufficient.

  5. Statements (1) and (2) TOGETHER are NOT sufficient.
Reveal answer Fill a bubble to check yourself
D Correct answer
Multiple choice

Four tangents drawn to a circle intersect each other to form a quadrilateral. The lengths of 3 consecutive sides are 6 cm, 10 cm and 16 cm. Find the length of the fourth side.

  1. 12 cm

  2. 9 cm

  3. 7 cm

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

For a quadrilateral circumscribed about a circle, the sums of opposite sides are equal. Let the sides be a, b, c, d in order. Then a + c = b + d. Here, 6 + 16 = 10 + d, so 22 = 10 + d, which means d = 12.

Multiple choice

Directions: The following problem consists of a question and three statements numbered I, II and III. You have to determine which of the following statements is/are necessary to answer the question. What will be the area of the given right-angled triangle? I. One of the angles is 20o more than other angle. II. The perimeter of the triangle is 3 times its base. III. The length of its hypotenuse is 5 cm.

  1. Only II and III

  2. Only III

  3. Either II or III

  4. Both I and III

  5. The question cannot be answered even with the information given in all three statements.
Reveal answer Fill a bubble to check yourself
A Correct answer
Multiple choice

A circle is drawn on the chord of a circle x2 + y2 = a2 as diameter. The chord lies on the line x + y = a. What is the equation of the circle?

  1. x2 + y2 - ax - ay + a2 = 0

  2. x2 + y2 - ax - ay = 0

  3. x2 + y2 + ax + ay = 0

  4. x2 + y2 + ax + ay - 2a2 = 0
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The chord of x^2 + y^2 = a^2 is x + y = a. The points of intersection are (a,0) and (0,a). A circle with this chord as diameter has the equation (x-a)(x-0) + (y-0)(y-a) = 0 => x^2 - ax + y^2 - ay = 0 => x^2 + y^2 - ax - ay = 0.

Multiple choice

A chord AB of a circle of radius 5.25 cm makes an angle of 60° at the centre of the circle. Find the area of the major and minor segments. (Take π = 3.14)

  1. 168 cm2

  2. 42 cm2

  3. 84 cm2

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

The area of the circle is pi * r^2 = 3.14 * 5.25^2 = 86.55 square centimeters. The area of the minor segment is calculated as the area of the 60-degree sector minus the area of the equilateral triangle OAB, which gives approximately 2.49 square centimeters. Subtracting this from the total area of the circle yields the area of the major segment as approximately 84.06 square centimeters, which matches option C.