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

AB is a diameter of a circle and PT is a tangent at point T to circle from an external point P. Find the value of PA if AB = 7 units and PT = 12 units. एक वृत्त का एक व्यास AB है और एक वाह्य बिन्दु P से वृत्त पर बिन्दु T पर एक स्पर्श रेखा PT है। PA का मान ज्ञात कीजिए यदि AB = 7 मात्रक और PT = 12 मात्रक है।

  1. 7

  2. 9

  3. 12

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

For any circle, if AB is a diameter and PT is a tangent from external point P, then PT² = PA × PB. This is the power of a point theorem. Given AB = 7, PT = 12. Let PA = x, then PB = PA + AB = x + 7. So: 12² = x(x + 7). 144 = x² + 7x. x² + 7x - 144 = 0. (x + 16)(x - 9) = 0. Therefore x = 9. Option A (7) would be the diameter itself, not PA.

Multiple choice

In a circle of radius 17 cm, two parallel chords of length 30 cm and 16 cm are drawn. If both the chords are on the same side of the centre, then the distance between the chords is 17 सेमी. की त्रिज्या वाले एक वृत्त में, 30 सेमी. और 16 सेमी. की दो समान्तर जीवायें खींची गयी हैं | यदि दोनों जीवायें केंद्र के एक ही ओर स्थित हैं, तो जीवाओं के बीच की दूरी है

  1. 9 cm./सेमी.

  2. 7 cm./सेमी.

  3. 23 cm./सेमी.

  4. 11 cm./सेमी.
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

For chord length L in a circle of radius R, perpendicular distance from center = sqrt(R² - (L/2)²). For 30 cm chord: distance = sqrt(289 - 225) = 8 cm. For 16 cm chord: distance = sqrt(289 - 64) = 15 cm. Since both chords are on same side of center, distance between them = 15 - 8 = 7 cm.

Multiple choice

In a cirlce of centre O, AB is a diameter and CD is a chord to the circle. ∠AOD = 106o then what is the value of ∠BCD. O वृत्त वाले केन्द्र में, AB एक व्यास है और CD वृत्त पर जीवा है। ∠AOD = 106o तो ∠BCD का मान क्या है?

  1. 53o

  2. 43o

  3. 40o

  4. 37o
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

AB is a diameter, so ∠ABC = 90° (angle in semicircle). Given ∠AOD = 106° (central angle), arc AD = 106°. ∠ABD = 106°/2 = 53° (inscribed angle on same arc). In right triangle ABC: ∠ABC + ∠BCD = 90°, so ∠BCD = 90° - 53° = 37°.

Multiple choice

A point P is 26 cm. away from the center O of a circle and PT is the tangent drawn from P to the circle is 10 cm., then OT is equal to - एक बिन्दु P एक वृत्त के केन्द्र से 26 सेमी. दूरी पर है और PT स्पर्श रेखा P से 10 सेमी. की खींची गई है। तो OT बराबर है-

  1. 18 cm./सेमी.

  2. 20 cm./सेमी.

  3. 24 cm./सेमी.

  4. 36 cm./सेमी.
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

By tangent-secant theorem: PT² = OP² - OT² (since OT ⟂ PT). Given OP = 26 cm and PT = 10 cm, we have OT² = 26² - 10² = 676 - 100 = 576. Thus OT = √576 = 24 cm. The radius is perpendicular to the tangent at the point of contact.

Multiple choice

\text{What is the distance between the orthocenter and the circumcentre of a triangle whose sides measure 30 cm, 34 cm and 16 cm?}

  1. 8 cm

  2. 15 cm

  3. 9 cm

  4. 7 cm

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

For a triangle with sides 30, 34, 16, check if right-angled: 16² + 30² = 256 + 900 = 1156, and 34² = 1156. Yes, it is right-angled with hypotenuse 34. In a right triangle, circumcenter is at the midpoint of hypotenuse (distance = 17 from right angle). Orthocenter is at the right angle vertex. Distance between them = 17 cm. None of 7, 8, 9, 15 match 17.

Multiple choice

AB is tangent to a circle, with center O and radius 8cm, at point M. If OA = 17 cm and OB = 10 cm, then what is the length of AB? AB एक वृत्त की स्पर्शरेखा है जिसका केंद्र O ,और त्रिज्या 8cm है को बिंदु M पर स्पर्श करता है । यदि OA = 17 सेमी और OB = 10 सेमी, तो AB की लंबाई क्या है?

  1. 25 cm/सेमी.

  2. 27 cm/सेमी.

  3. 21 cm/सेमी.

  4. 17 cm/सेमी.
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Since OM is perpendicular to tangent AB at M, triangles OMA and OMB are right triangles. Using Pythagoras: AM = √(OA² - OM²) = √(289 - 64) = 15, BM = √(OB² - OM²) = √(100 - 64) = 6. So AB = 15 + 6 = 21 cm (assuming A and B are on opposite sides of M).

Multiple choice

In a circle, O is the center and PQ is any chord and M is the point on the major arc. If angle ∠ PMQ = 75o then find the value of angle ∠OQP. एक गोले में O केंद्र तथा PQ एक जीवा है तथा M ऋजुचाप पर एक बिंदु है । यदि कोण ∠ PMQ = 75o है, तो कोण ∠OQP का मान ज्ञात कीजिये ।

  1. 15o

  2. 150o

  3. 75o

  4. 30o
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Angle PMQ is an inscribed angle subtending arc PQ. Since M is on the major arc, angle POQ (central angle) = 2 × 75° = 150°. In triangle OPQ (isosceles with OP = OQ), angle OQP = (180° - 150°)/2 = 15°.

Multiple choice

AB and CD are two parallel chords of a circle of lengths 16 cm and 12 cm respectively. If the chords are on the same side of the centre and the distance between them is 2 cm, then find the radius of the circle ? AB और CD, एक वृत्त की दो समांतर जीवाएं हैं, जिनकी लम्बाई क्रमश: 16 सेमी और 12 सेमी है। यदि जीवाएं केंद्र की समान ओर हैं एवं इनके मध्य की दूरी 2 सेमी है तो वृत्त की त्रिज्या ज्ञात कीजिए।

  1. 10 cm/सेमी.

  2. 12 cm/सेमी.

  3. 5 cm/सेमी.

  4. 20 cm/सेमी.
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

For chord AB (16 cm), perpendicular distance from center = √(r² - 64) = d1. For chord CD (12 cm), perpendicular distance = √(r² - 36) = d2. Given: d1 - d2 = 2 cm. Substituting r = 10 cm: d1 = √(100 - 64) = 6 cm, d2 = √(100 - 36) = 8 cm. Indeed 8 - 6 = 2 cm, which matches. The radius is 10 cm.

Multiple choice

The length of arc whose central angle is θ is 26.4 cm and radius of circle is 28 cm. Find the θ = ? किसी चाप की लम्बाई 26.4 सेमी है जो वृत्त के केन्द्र पर θ कोण बनाता है। यदि वृत्त की त्रिज्या 28 सेमी है तो θ का मान = ?

  1. 60o

  2. 75o

  3. 64o

  4. 54o
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Arc length formula: s = rθ (θ in radians). Given s = 26.4 cm, r = 28 cm, so θ = 26.4/28 radians. Converting to degrees: θ = (26.4/28) × (180/π) ≈ (0.942857) × 57.296° ≈ 54°. The conversion factor from radians to degrees is 180/π ≈ 57.3°.

Multiple choice

An isosceles triangle is formed by the two radii in a circle of radius 14 cm and the length of the chord connecting the two points is 8√10 cm. What is the area of the part of circle excluding the isosceles triangle? (APPROXIMATE) 14 से.मी. त्रिज्या के एक वृत्त में दो त्रिज्या का उपयोग करके एक समद्विबाहु त्रिकोण बनाया गया है और दो बिंदुओं को जोड़ने वाली जीवा की लंबाई 8√10 से.मी है। तो समद्विबाहु त्रिभुज को छोड़कर, वृत्तीय भाग का क्षेत्रफल कितना है?(लगभग)

  1. 440.8 cm2/ 440.8 सेमी2

  2. 339.8 cm2/ 339.8 सेमी2

  3. 539.2 cm2/ 539.2 सेमी2

  4. 500.2 cm2/ 500.2 सेमी2
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Circle area = πr² = π×14² = 196π ≈ 615.8 cm². Isosceles triangle with two sides = 14 and base = 8√10. Using cosine law: cos(angle) = (14² + 14² - (8√10)²) / (2×14×14) = (392 - 640) / 392 = -248/392 ≈ -0.633. Height from vertex = 14×sin(angle/2). Triangle area ≈ 76.6 cm². Remaining area = 615.8 - 76.6 ≈ 539.2 cm².

Multiple choice

Chords AB and CD of a circle intersect externally at P. If AB = 6 cm, CD = 3 cm and PD = 5 cm, then the length of PB is- एक वृत्त की दो जीवा AB तथा CD एक दूसरे को वृत्त के बाहर बिन्दु P पर प्रतिच्छेदित करती है। यदि AB= 6 सेमी CD= 3 सेमी तथा PD= 5 सेमी. हो तब PB की लम्बाई ज्ञात कीजिए?

  1. 5 cm/सेमी

  2. 7.35 cm/सेमी

  3. 6 cm/सेमी

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

For chords intersecting externally at P: PA × PB = PC × PD. With AB = 6, let PB = x, so PA = x + 6. With CD = 3 and PD = 5, PC = 5 + 3 = 8. Then (x + 6)x = 8 × 5 = 48, giving x² + 6x - 48 = 0. Factoring (x + 12)(x - 4) = 0, so x = 4 (positive length). PB = 4 cm.

Multiple choice

The Perimeters of two similar triangles are 30 cm and 20 cm respectively. If one side of the first triangle is 9 cm determine the corresponding side of the second triangle. दो समरूप त्रिभुजों की परिमाप क्रमशः 30 सेमी और 20 सेमी है। यदि पहले त्रिभुज की एक भुजा 9 सेमी लम्बी है। तो दूसरे त्रिभुज की भुजा की लम्बाई ज्ञात कीजिए ?

  1. 15 cm/सेमी

  2. 6 cm/सेमी

  3. 13.5 6 cm/सेमी

  4. 5 cm/सेमी
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Similar triangles have proportional sides and perimeters. The ratio of perimeters 30:20 simplifies to 3:2, so all corresponding sides are in the same ratio. If the first triangle's side is 9 cm, the corresponding side in the second triangle is 9 × (2/3) = 6 cm.

Multiple choice

The area of the square inscribed in a circle of radius 12 cm is/ 12 सेमी त्रिज्या के एक वृत्त में बने वर्ग का क्षेत्रफल है-

  1. 288 cm2

  2. 225 cm2

  3. 256 cm2

  4. 320 cm2
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

For a square inscribed in a circle of radius 12 cm, the diagonal of the square equals the diameter of the circle, which is 24 cm. If side is s, then diagonal = s√2, so s = 24/√2 = 12√2 cm. Area = s² = (12√2)² = 144 × 2 = 288 sq cm.

Multiple choice

If distance between orthocentre and circumcentre of a triangle is 6 cm then what is the distance between its centroid and circumcentre? यदि एक त्रिभुज के लम्ब केन्द्र और परिकेन्द्र के मध्य अन्तर 6 सेमी. है तो इसके केन्द्रक और परिकेन्द्र के मध्य की दूरी क्या है।

  1. 4 cm/सेमी.

  2. 2 cm/सेमी.

  3. 37 cm/सेमी.

  4. 35 cm/सेमी.
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

In any triangle, the centroid (G), circumcenter (O), and orthocenter (H) lie on the Euler line. The centroid divides the line segment joining the orthocenter and circumcenter in the ratio 2:1, with the centroid being closer to the circumcenter. If OH = 6 cm, then OG = (1/3) × OH = 6/3 = 2 cm. Option B matches. Options A, C, and D don't satisfy the Euler line property.