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

In a circle of radius 10 cm, with centre O, PQ and PR are two chords each of length 12 cm. PO intersects chord QR at the point S. The length of OS is

  1. 2.8 cm

  2. 2.5 cm

  3. 3.2 cm

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

In a circle, the perpendicular from the center to a chord bisects the chord. QR is a chord. Since PQ=PR=12, triangle PQR is isosceles. PO is the perpendicular bisector of QR. Using Pythagoras in triangle OMQ (where M is midpoint of PQ), find the distance.

Multiple choice

The number of common tangents to the circles x2 + y2 – 4x – 6y – 12 = 0 and x2 + y2 + 6x + 18y + 26 = 0, is

  1. 1

  2. 2

  3. 3

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

Circle 1: center (2, 3), radius = sqrt(4+9+12) = 5. Circle 2: center (-3, -9), radius = sqrt(9+81-26) = 8. Distance between centers = sqrt((2 - -3)^2 + (3 - -9)^2) = sqrt(5^2 + 12^2) = 13. Since distance (13) = r1 + r2 (5 + 8), the circles touch externally. They have 3 common tangents.

Multiple choice

Direction: The following question has two statements, (1) and (2). Answer the question using the following options: A parallelogram is inscribed in a circle. What is the perimeter of the parallelogram? (1) One side of the parallelogram is 5 cm. (2) The area of the circle is 169 π cm2.

  1. if statement (1) alone is sufficient but statement (2) alone is not

  2. if statement (2) alone is sufficient but statement (1) alone is not

  3. if both (1) and (2) together are sufficient but none of them alone is sufficient

  4. if both statements are sufficient independently

  5. if both (1) and (2) together are not sufficient
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

A parallelogram inscribed in a circle must be a rectangle. The diagonal of the rectangle is the diameter of the circle. Statement (2) gives the area of the circle (pi * r^2 = 169 * pi), so r = 13, diameter = 26. With one side = 5, the other side is sqrt(26^2 - 5^2) = sqrt(676 - 25) = sqrt(651). Perimeter = 2 * (5 + sqrt(651)). Both statements are required.

Multiple choice

If G is the centroid of ?ABC and ?ABC = 48 cm2, then the area of ?BGC is

  1. 8 cm2

  2. 16 cm2

  3. 24 cm2

  4. 32 cm2
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Centroid divides a triangle into 3 smaller triangles of equal area. Each of triangles BGC, CGA, AGB has area = (1/3) × area of ABC = 48/3 = 16 cm². This is because all three have equal base lengths on the sides of ABC and share the same height from G.

Multiple choice

If two concentric circles are of radii 5 cm and 3 cm, then the length of the chord of the larger circle which touches the smaller circle is-

  1. 6 cm

  2. 7 cm

  3. 10 cm

  4. 8 cm
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The chord of the larger circle (radius 5 cm) that touches the smaller circle (radius 3 cm) is tangent to it. The distance from the center to the chord equals the smaller radius (3 cm). Using the chord length formula: 2√(r² - d²) = 2√(25 - 9) = 2√16 = 2(4) = 8 cm.

Multiple choice

In ?ABC, M is the mid point of BC. Length of AM is 9. N is a point on AM such that MN = 1. Distance of N from the centroid of ?ABC is equal to :

  1. 3

  2. 6

  3. 5

  4. 2

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

In a triangle, the centroid divides the median in ratio 2:1 (from vertex). AM is a median of length 9. The centroid is at distance (2/3) × 9 = 6 from A. Point N is on AM with MN = 1, so N is at distance 9 - 1 = 8 from A. Distance from N to centroid = 8 - 6 = 2.

Multiple choice

In a triangle ABC, O is its circumcentre and ∠BAC = 500.The measure of ∠OBC is.

  1. 300

  2. 400

  3. 500

  4. 600
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

In any triangle, the angle at the center (circumcentre) is twice the angle at the circumference subtended by the same arc. Given ∠BAC = 50°, the central angle ∠BOC = 2 × 50° = 100°. Since O is the circumcentre, OB = OC (both are radii), so triangle OBC is isosceles. Therefore, ∠OBC = (180° - 100°)/2 = 40°. This uses the central angle theorem and properties of isosceles triangles.