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

A, B, C, D, E ........ Z are the points marked on the circumference of a circle equidistantly. What can be the maximum number of triangles which can be formed using three points as vertices such that their circumcentre lies on one of the sides of a triangle?

  1. 24

  2. 13

  3. 372

  4. 312

  5. 624
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

For a triangle inscribed in a circle, the circumcenter lies on a side if and only if the triangle is a right-angled triangle. A right-angled triangle is formed by choosing the diameter as one side and any other point on the circle as the third vertex. There are 26 points, so 13 diameters. For each diameter, there are 26 - 2 = 24 points to choose from. Total = 13 * 24 = 312.

Multiple choice

The tangents drawn from the origin to the circle x2 + y2 – 14x + 2y + 25 = 0 are

  1. Parallel

  2. Perpendicular

  3. Coincident

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

The circle is x^2 + y^2 - 14x + 2y + 25 = 0. The origin (0,0) is outside the circle (0+0-0+0+25 > 0). Tangents from a point to a circle are perpendicular if the point lies on the director circle. Director circle is x^2 + y^2 - 14x + 2y + 25 - r^2 = 0. r^2 = g^2 + f^2 - c = 49 + 1 - 25 = 25. Director circle: x^2 + y^2 - 14x + 2y = 0. Since (0,0) satisfies this, the tangents are perpendicular.

Multiple choice

The length of the tangent from (x1, y1) to the circle x2 + y2 + 2gx + 2fy + c = 0, is

  1. (x12 + y12 + 2gx1 + 2fy1 + c)

  2. (x12 + y12)1/2

  3. [(x1 + g)2 + (y1 + f)2]1/2

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

The length of the tangent from a point (x1, y1) to a circle x^2 + y^2 + 2gx + 2fy + c = 0 is given by the square root of the power of the point, which is sqrt(x1^2 + y1^2 + 2gx1 + 2fy1 + c). Option A lacks the square root, and other options are incorrect.

Multiple choice

A circle is inscribed in a given square and another circle is circumscribed about the square. What is the ratio of the area of the inscribed circle to that of the circumscribed circle?

  1. 2 : 3

  2. 3 : 4

  3. 1 : 4

  4. 1 : 2
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Let square side = a. Inscribed circle radius = a/2. Area = pi*(a/2)^2 = pi*a^2/4. Circumscribed circle radius = diagonal/2 = (a*sqrt(2))/2 = a/sqrt(2). Area = pi*(a/sqrt(2))^2 = pi*a^2/2. Ratio = (pi*a^2/4) / (pi*a^2/2) = 1/2.

Multiple choice

If the area of an equilateral triangle is approximately 68 cm2, what is the area of the triangle whose sides are equal in length to the medians of the equilateral triangle?

  1. 51 cm2

  2. 45 cm2

  3. 60 cm2

  4. Cannot be determined
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The medians of an equilateral triangle with side length 's' have length (sqrt(3)/2) * s. The area of a triangle with these medians as sides is 3/4 of the original area, but the question asks for the area of a triangle whose sides are equal in length to the medians, which is 3/4 of the original area. 68 * 0.75 = 51.

Multiple choice

O is the circumcentre of a triangle ABC with circumradius 13 cm. If BC = 24 cm and OD is perpendicular to BC, then the length of OD is

  1. 3 cm

  2. 4 cm

  3. 5 cm

  4. 7 cm
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Since O is the circumcentre of triangle ABC, the distance from O to any vertex is equal to the circumradius, so OB = 13 cm. The perpendicular OD from the center O to the chord BC bisects the chord, meaning BD = 12 cm. Applying the Pythagorean theorem in the right-angled triangle ODB, we get OD^2 + 12^2 = 13^2, which simplifies to OD = 5 cm.