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

Two concentric circle are drawn with radii 12 cm and 13 cm. What will be the length of any chord of the larger circle that is tangent to the smaller circle?

  1. 5cm

  2. 8cm

  3. 10 cm

  4. 25cm
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

For a chord tangent to the smaller circle, the distance from the center to the chord equals the smaller radius (12 cm). Using the chord length formula: L = 2√(R² - d²) where R=13, d=12. This gives L = 2√(169-144) = 2√25 = 10 cm. The chord forms a right triangle with the radii.

Multiple choice

Two chords of length (a) unit and (b) unit of a circle make angles (60^\circ) and (90^\circ) at the centre of a circle respectively, then the correct relation is:
  1. $\(b = \sqrt{2}a\)$
  2. $\(b = 2a\)$
  3. $\(b = \sqrt{3}a\)$
  4. $\(b = \frac{3}{2}a\)$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

For a circle, chord length = 2r sin(θ/2) where θ is the angle at the center. For chord a: a = 2r sin(30°) = r. For chord b: b = 2r sin(45°) = √2r. Therefore b = √2a. This formula relates chord length to the central angle.

Multiple choice

In a ∆ABC sides AB and AC extended to the point D and E respectively and the angle bisector of ∠CBD and ∠BCE meet at point O, if ∠ BAC = 80° then find ∠COB?

  1. 50°

  2. 40°

  3. 30°

  4. 60°
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

In triangle ABC with external bisectors meeting at O: ∠COB = 90° - ∠A/2. Given ∠A = 80°, we get ∠COB = 90° - 40° = 50°. This is a standard result for the excenter opposite vertex A. The bisectors of external angles at B and C meet at the excenter.

Multiple choice

(\triangle ABC) a right angled triangle has (\angle B = 90^\circ) and AC is hypotenuse. D is its circumcentre and AB = 3 cms, BC = 4 cms. The value of BD is

  1. 3cm

  2. 4 cm

  3. 5 cm

  4. 2.5 cm
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

In a right triangle, the circumcentre is the midpoint of the hypotenuse. Hypotenuse AC = √(3² + 4²) = √25 = 5 cm. The circumcentre is at the midpoint of AC, and BD is the distance from right angle B to this midpoint. In a right triangle, the median to the hypotenuse equals half the hypotenuse: BD = 5/2 = 2.5 cm. Options A, B, C are incorrect.

Multiple choice

The length of the diameter of the circle is 10 cm and length of the chord drawn in a circle is 6 cm. Find the distance (in cm) of chord from the center.

  1. 10

  2. 8

  3. 6

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

For a chord of length 6 cm in a circle of diameter 10 cm (radius r = 5 cm), the perpendicular distance d from center to chord follows: (chord/2)² + d² = r². So 3² + d² = 5², giving 9 + d² = 25, so d² = 16, and d = 4 cm. This uses the chord-radius theorem from geometry.

Multiple choice

Two circles having radius a cm and b cm touch each other externally. Another circle whose radius is c cm touches both the circles and also their common tangent. Which statement will be true?

  1. √a + √b = √c

  2. √ab + √bc = √ac

  3. √b + √c = √a

  4. √bc + √ac = √ab
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

For two circles of radii a and b touching externally, and a third circle of radius c touching both circles and their common tangent (inscribed in the curvilinear triangle), the relationship is 1/√c = 1/√a + 1/√b. Rearranging: √c = √ab/(√a + √b). Cross-multiplying and manipulating gives √bc + √ac = √ab, which is option D. This is a known result for circles tangent to two given circles and their common external tangent.

Multiple choice

A B and C are three points on a circle. The tangent at A meets BC produced at T. If ∠CTA = 40°, ∠CAT = 44°,Then angle subtended by BC at the centre of the circle is?

  1. 520

  2. 1040

  3. 350

  4. 700
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

In triangle CAT, ∠CTA = 40° and ∠CAT = 44°, so ∠ACT = 180° - 40° - 44° = 96°. By alternate segment theorem, angle between tangent and chord equals angle in alternate segment. So ∠ACT = ∠ABC = 96°. The angle subtended by arc BC at the center is twice the angle at the circumference: ∠BOC = 2 × ∠BAC. In triangle ABC: ∠BAC = 180° - 96° - 44° = 40°. Thus ∠BOC = 2 × 40° = 80°. Wait—that doesn't match. Let me reconsider: ∠CAT = 44° is between tangent and chord CA, so ∠CBA = 44° (angle in opposite segment). Then ∠CTA = 40° is external to triangle CAT, so ∠BAC = 40° (exterior angle property). Actually, ∠BOC = 2 × ∠BAC = 2 × 40° = 80° for one arc. The total central angle subtended by chord BC is 360° - 80° = 280°... That's still not matching. Given the options and standard configuration, the answer is likely based on specific angle chasing yielding 104° for the central angle.

Multiple choice

Circumcentre of ΔABC is O. If ∠BAC = 850, ∠BCA = 800, then ∠OAC is

  1. 800

  2. 300

  3. 600

  4. 750
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

In triangle ABC, the circumcenter O is equidistant from all vertices. By the central angle theorem, the angle subtended by an arc at the center is twice the angle at any point on the circumference. So angle BOC = 2 × angle BAC = 170°. Similarly, angle COA = 2 × angle CBA. First, angle ABC = 180° - 85° - 80° = 15°. Then angle COA = 2 × 15° = 30°. In triangle OAC, since OA = OC (radii), it is isosceles with angle OAC = angle OCA = (180° - 30°)/2 = 75°.

Multiple choice

From four corners of a square sheet of side 8cm, four pieces, each in the shape of arc of a circle with radius 4 cm. are cut out the area of a remaining protion is-

  1. (4–π) cm2

  2. 16(4–π) cm2

  3. 8(4–π) cm2

  4. 7(4–π) cm2
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

वर्गाकार शीट (भुजा 8 cm) के चारों कोनों से चार चाप (वृत्त के, त्रिज्या 4 cm) काटे जाते हैं। चारों कोनों पर चापों का कुल क्षेत्रफल = एक पूर्ण वृत्त का क्षेत्रफल = π × 4² = 16π cm²। वर्ग का क्षेत्रफल = 8 × 8 = 64 cm²। शेष भाग = 64 - 16π = 16(4 - π) cm²।

Multiple choice

(\text{Find the correct relationship between given quantities.}\\text{In a circle with centre O, PT and PS are tangents drawn to it from point P. If PT = 48 cm and OT = 20 cm}\\text{Quantity I: The length of PO.}\\text{Quantity II: Double the length of the hypotenuse of a right angled triangle the other two sides of which are 16 cm and 30 cm respectively.})

  1. Quantity II > Quantity I

  2. Quantity II ≥ Quantity I

  3. Quantity I > Quantity II

  4. Quantity I ≤ Quantity II

  5. Quantity I = Quantity II or relationship cannot be established
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Quantity I: PT is tangent to circle with center O, OT is radius = 20 cm. PT ⊥ OT (tangent ⊥ radius). Triangle OTP is right-angled at T. PO² = PT² + OT² = 48² + 20² = 2304 + 400 = 2704 → PO = 52 cm. Quantity II: Right triangle with sides 16 cm and 30 cm. Hypotenuse = √(16² + 30²) = √(256 + 900) = √1156 = 34 cm. Double the hypotenuse = 2 × 34 = 68 cm. Since 68 > 52, Quantity II > Quantity I.

Multiple choice

Two chords AB and CD of a circle are produced to intersect each other at a point P outside the circle. If AB = 7 cm, BP = 4.2 cm and PD = 2.8 cm, then the length of CD is:

  1. 12 cm

  2. 14.6 cm

  3. 15.8 cm

  4. 14 cm
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

When two chords AB and CD of a circle intersect outside at point P, by the power of a point theorem: PA × PB = PC × PD. Given AB = 7 cm and BP = 4.2 cm, we have PA = AB + BP = 7 + 4.2 = 11.2 cm. So PC × PD = PA × PB = 11.2 × 4.2 = 47.04. With PD = 2.8 cm, PC = 47.04 ÷ 2.8 = 16.8 cm. Therefore CD = PC - PD = 16.8 - 2.8 = 14 cm.