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 y = 2x is a chord of the circle x2 + y2 = 10x, then equation of the circle whose diameter is the chord is

  1. x2 + y2 + 2x + 4y = 0

  2. x2 + y2 + 2x – 4y = 0

  3. x2 + y2 – 2x – 4y = 0

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

The equation of a circle with chord y = 2x and circle x^2 + y^2 - 10x = 0 is given by S + kL = 0, where S is the circle and L is the line. Substituting the line into the circle equation gives the intersection points, but the standard formula for a circle with a given chord as diameter is S + lambda(L) = 0. Solving for the specific circle yields x^2 + y^2 - 2x - 4y = 0.

Multiple choice

Two triangles are similar but not congruent and the lengths of the sides of the first are 6cm, 11cm and 12cm. The ratio of corresponding sides of first and second triangle is 1:2 What is the perimeter of the second triangle:

  1. 29cm

  2. 53cm

  3. 58cm

  4. 56cm
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The perimeter of the first triangle is 6 + 11 + 12 = 29cm. Since the ratio of sides is 1:2, the ratio of perimeters is also 1:2. Therefore, the perimeter of the second triangle is 29 * 2 = 58cm.

Multiple choice

Two concentric circles are of radii 13 cm and 5 cm. The length of the chord of larger circle which touches the smaller circle is

  1. 24 cm

  2. 20 cm

  3. 12 cm

  4. 26 cm
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The radius of the smaller circle (5 cm) is the perpendicular distance from the center to the chord. This forms a right triangle with the radius of the larger circle (13 cm) as the hypotenuse. Half-chord length = sqrt(13^2 - 5^2) = sqrt(169 - 25) = sqrt(144) = 12 cm. Total chord length = 2 * 12 = 24 cm.

Multiple choice

At one end P of a diameter PQ of a circle of radius 5 cm, tangent XPY is drawn. The length of the chord RS parallel to XY and at a distance of 8 cm from P is :

  1. 7 cm

  2. 5 cm

  3. 8 cm

  4. 6 cm
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The circle has radius 5, so the diameter is 10. The chord RS is parallel to the tangent at P, meaning it is perpendicular to the diameter. The distance from P to the chord is 8. The distance from the center to the chord is |8 - 5| = 3. Using the Pythagorean theorem on the triangle formed by the radius, the distance to the chord, and half the chord length: (half-chord)^2 + 3^2 = 5^2. (half-chord)^2 = 25 - 9 = 16. Half-chord = 4. Total chord length = 8.

Multiple choice

To construct a pair of tangents to a circle at an angle of 60° to each other, it is needed to draw tangents at endpoints of those two radii of the circle, the angle between them should be:

  1. 1000

  2. 900

  3. 800

  4. 1200
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The tangents to a circle from an external point form a quadrilateral with the two radii. The angles between the radii and the tangents are 90 degrees. If the angle between tangents is 60 degrees, the angle between the radii must be 180 - 60 = 120 degrees.

Multiple choice

If two circles, each of radius 5 units, touch each other at (1, 2) and the equation of their common tangent is 4x + 3y = 10, then the equation of the circle, a portion of which lies in all the quadrants, is

  1. x2 + y2 – 10x – 10y + 25 = 0

  2. x2 + y2 + 6x + 2y – 15 = 0

  3. x2 + y2 + 2x + 6y – 15 = 0

  4. x2 + y2 + 10x + 10y + 25 = 0
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The center of the circles must lie on a line perpendicular to the tangent 4x+3y=10, passing through (1,2). The line is 3x-4y+c=0. 3(1)-4(2)+c=0 => c=5. The centers are at distance 5 from (1,2) along this line. The equations provided in the options can be checked for the center and radius.

Multiple choice

A tangent is drawn to a circle from an external point p. What is the length of the tangent? (1) The radius of the circle is given (2) The distance between the center and the point p is given.

  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
C Correct answer
Explanation

Let r be the radius and d be the distance from the center to point P. The tangent, radius, and the line from center to P form a right triangle. The length of the tangent is sqrt(d^2 - r^2). Both r and d are needed to calculate this.

Multiple choice

The lengths of two adjacent sides of a rectangle inscribed in a circle are 5 centimeters and 12 centimeters. Quantity A Radius of the circle Quantity B 5.5 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
A Correct answer
Explanation

A rectangle inscribed in a circle has a diagonal equal to the circle's diameter. Using the Pythagorean theorem, the diagonal is sqrt(5^2 + 12^2) = sqrt(25 + 144) = sqrt(169) = 13. The radius is half the diameter, so r = 13/2 = 6.5. Since 6.5 > 5.5, Quantity A is greater.