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

The tangents are drawn at the extremities of a diameter AB of a circle with centre P. If a tangent to the circle at the point C intersects the other two tangets at Q and R, then the measure of the ∠QPR is- P केन्द्र वाले एक वृत्त का व्यास AB के छोरों पर स्पर्श रेखाएं खींची गई हैं। यदि बिंदु C पर वृत्त की एक स्पर्श रेखाओं का Q तथा R पर प्रतिच्छेदित करें तो ∠QPR का माप है-

  1. 450

  2. 600

  3. 900

  4. 1800
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Tangents at endpoints of diameter are parallel (both perpendicular to radius). The tangent at point C is perpendicular to radius PC. In quadrilateral QPRC, angle between parallel tangents is 0. By properties of tangents from external points, angle QPR = 90 degrees. This is a standard property: tangents at diameter endpoints form a right angle with any other tangent.

Multiple choice

AB is a chord to a circle and PAT is the tangent to the circle at A. If ∠ BAT = 750 and ∠ BAC = 450, C being a point on the circle, then ∠ ABC is equal to- AB एक वृत्त की जीवा है और PAT वृत्त की A पर स्पर्श रेखा है। यदि ∠ BAT = 750 और ∠ BAC = 450, C वृत्त एक बिन्दु है, तो ∠ ABC है

  1. 400

  2. 450

  3. 600

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

By tangent-chord theorem: angle between tangent and chord equals angle in alternate segment. So angle BAT = angle BCA = 75. In triangle ABC: angle BAC + angle ABC + angle BCA = 180. 45 + angle ABC + 75 = 180. Angle ABC = 180 - 120 = 60 degrees. Use alternate segment theorem to relate angles.

Multiple choice

If AB and AC are two tangents to a circle, ST is another tangent to the circle at the point P on the circle, which intersects AB and AC at S and T respectively. If AB = 15 cm. and TA = 9 cm, then PT is- यदि AB और AC एक वृत्त की दो स्पर्श-रेखाएं हैं, वृत्त पर स्थित बिन्दु P पर ST एक दूसरी स्पर्श-रेशा है, जो AB और AC को क्रमशः बिन्दु S और T पर काटती है । यदि AB = 15 सेमी. और TA = 9 सेमी. तो PT है-

  1. 4.5 cm./सेमी.

  2. 6.5 cm./सेमी.

  3. 6 cm./सेमी.

  4. 7.5 cm./सेमी.
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Since AB and AC are tangents from A, AB = AC = 15 cm (tangent segments from same point are equal). Given TA = 9 cm, then TC = AC - TA = 15 - 9 = 6 cm. By tangent segment theorem, PT = TC = 6 cm (tangents from T to the circle). Options A, B, and D are incorrect.

Multiple choice

Two circles touch each other externally at C and AB is a common tangent to the circle. Then, ∠ACB is - दो वृत्त एक दूसरे को वाह्यतः C पर स्पर्श करते है और A B वृत्त की एक उभयनिष्ठ स्पर्श रेखा है।∠ACB है-

  1. equal to 90o

  2. less than 90o

  3. greater than 90o

  4. greater than 120o
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

For two circles touching externally at C with common tangent AB, draw radii from centers to points of tangency. The radii to tangent points are perpendicular to AB. This creates two right angles. The triangle formed is such that ∠ACB = 90° because the radii to C are collinear (since circles touch at C), making the angle subtended by AB at C equal to 90° by tangent properties. Options B, C, D are incorrect.

Multiple choice

Two arcs of two different circles are of equal lengths. If these arcs subtends angles of 300 and 450 at the centres of the circles. Find the ratio of the radii of the two circles. दो भिन्न वृत्तों के दो वृत्त-खण्ड समान लम्बाई के हैं। यदि ये वृत्त के केन्द्रों पर क्रमशः 300 और 450 का कोण अन्तरित करते हैं। वृत्तों की त्रिज्याओं में अनुपात ज्ञात कीजिए।

  1. 4 : 3

  2. 2 :1

  3. 3 : 2

  4. 2 : 3
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Arc length L = r × θ (θ in radians). For equal arcs: r₁×θ₁=r₂×θ₂. Given θ₁=300°=300×π/180=5π/3 radians, θ₂=450°=450×π/180=5π/2 radians. Ratio r₁/r₂=θ₂/θ₁=(5π/2)/(5π/3)=3/2. So r₁:r₂=3:2.

Multiple choice

The ratio between radius of two concentric circles are 5 : 3. Find the ratio between length of tangent of smaller circle whose ends are at the cirfcrcumference of the largest circle to the diameter of smaller circle. दो संकेन्द्रीय वृत्तों की त्रिज्याओं का अनुपात 5 : 3 है तो छोटे वृत्त पर स्पर्श रेखा जिसके अंत बिंदु बड़े वृत्त के परिधि पर है तथा छोटे वृत्त के व्यास के बीच का अनुपात ज्ञात कीजिए |

  1. 3:4

  2. 4:3

  3. 2:3

  4. 3:2
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

For concentric circles with radii 5 and 3, using Pythagoras theorem on the right triangle formed by the radius of larger circle (5), radius of smaller circle (3), and half the tangent: tangent² = 5² - 3² = 16, so tangent length = 4. The diameter of smaller circle = 2×3 = 6. Therefore, ratio of tangent to diameter = 4:6 = 4:3 in simplified form.

Multiple choice

Find the area of rectangle which is twice of the area of square inscribed in a circle of radius 14 cm. उस आयत का क्षेत्रफल ज्ञात कीजिए जिसका क्षेत्रफल उस वर्ग के क्षेत्रफल के दुगुने के बराबर है जो 14 सेमी त्रिज्या वाले वृत्त के अंतर्गत बना हुआ है।

  1. 784 sq.cm/वर्ग सेमी.

  2. 724 sq.cm/वर्ग सेमी.

  3. 789 sq.cm/वर्ग सेमी.

  4. 800 sq.cm/वर्ग सेमी.
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

For a circle of radius 14 cm, the square's diagonal equals the circle's diameter (28 cm). If square's side is 'a', then a√2 = 28, so a = 28/√2 = 19.8 cm. Area of square = a² = 392 sq.cm. The rectangle's area is twice this: 2 × 392 = 784 sq.cm.

Multiple choice

AB is a diameter of a circle with centre at O. DC is a chord such that DC || AB. If ∠BAC=30o, then find ∠ADC? O केंद्र वाले वृत्त का व्यासAB है। DC उसकी जीवा है और DC || AB यदि ∠BAC=30o हो तो ∠ADC का मान ज्ञात कीजिए?

  1. 120o

  2. 110o

  3. 115o

  4. 100o
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Since AB is a diameter, ∠ACB = 90° (angle in a semicircle). In ΔABC, ∠ABC = 180° - 90° - 30° = 60°. ABCD is a cyclic quadrilateral with DC || AB. Opposite angles in a cyclic quadrilateral sum to 180°, so ∠ADC = 180° - ∠ABC = 180° - 60° = 120°.

Multiple choice

ABC is a right angled triangle with BC = 6 cm. and AB = 8 cm. A circle with centre O is inscribed in Δ ABC. Find the radius of the circle. ABC एक समकोण त्रिभुज है जिसमें BC = 6 सेमी. और AB = 8 सेमी. है, त्रिभुज ABC में एक अंतः वृत्त O केन्द्र का है, तो उस वृत्त की त्रिज्या ज्ञात कीजिए।

  1. 1 cm./सेमी.

  2. 2 cm./सेमी.

  3. 3 cm./सेमी.

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

In a right triangle with legs 6 and 8, the hypotenuse is 10 (6-8-10 Pythagorean triple). For an incircle in a right triangle, r = (a + b - c)/2 = (6 + 8 - 10)/2 = 2 cm. Alternatively, area = (1/2) × 6 × 8 = 24, semiperimeter = 12, and r = Area/s = 24/12 = 2.

Multiple choice

Two parallel chords of a circle of diameter 20 cm are 12 cm and 16 cm long. If the chords are in the same side of the centre, then the distance between them is एक 20 सेमी. व्यास वाले वृत्त में दो समान्तर जीवायें क्रमशः 12 सेमी. और 16 सेमी. हैं | यदि जीवायें केंद्र के एक ही तरफ स्थित हैं, तो उनके बीच की दूरी है

  1. 28 cm./सेमी.

  2. 2 cm./सेमी.

  3. 4 cm./सेमी.

  4. 8 cm./सेमी.
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

For a circle of diameter 20 cm, radius = 10 cm. For chord of length 12 cm: distance from center = √(10² - 6²) = √(100 - 36) = √64 = 8 cm. For chord of length 16 cm: distance from center = √(10² - 8²) = √(100 - 64) = √36 = 6 cm. Since chords are on same side of center, distance between them = 8 - 6 = 2 cm. Verify: Both distances are less than radius (10 cm), so chords exist and are on same side ✓.

Multiple choice

A chord of length 30 cm is at a distance of 8 cm from the centre of a circle. The radius of the circle is : 30 सेमी लम्बी एक जीवा, एक वृत्त के केन्द्र से 8 सेमी दूर है। तदनुसार, उस वृत्त की त्रिज्या कितनी है?

  1. 23 cm/सेमी

  2. 21 cm/सेमी

  3. 19 cm/सेमी

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

In a circle, a chord of length L at distance d from center forms a right triangle with radius as hypotenuse. By Pythagorean theorem: r² = d² + (L/2)². Here: r² = 8² + (30/2)² = 64 + 225 = 289. So r = √289 = 17 cm.

Multiple choice

Two tangents are drawn from a point P to a circle at A and B .O is the center of circle .If ∠ AOP = 600,then ∠ APB is - एक बिंदु P से एक वृत्त के बिंदु A और B पर दो स्पर्श रेखा खीचीं गई है। O वृत्त का केद्र है। यदि ∠AOP = 600 हो, तो ∠ APB है -

  1. 300

  2. 600

  3. 900

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

The tangent is perpendicular to the radius at the point of contact. In right triangle AOP with angle AOP = 60°, angle APO = 30°. Similarly, angle BPO = 30°. Therefore angle APB = 30° + 30° = 60°. The value 600 in the question is a typo for 60°.

Multiple choice

If radii of two circles are respectively 11 cm and 13 cm and length of transverse common tangent to the two circle is 7 cm, then what is the distance between centres of the two circles? यदि दो वृत्तों की त्रिज्याएं क्रमशः 11 सेमी. और 13 सेमी. है और दोनों वृत्तों की उभयनिष्ठ अनुप्रस्थ स्पर्श रेखा 7 सेमी. है, तो दोनों वृत्तों के केन्द्रों के बीच की दूरी क्या है?

  1. 24 cm/सेमी

  2. 25 cm/सेमी

  3. 27cm/सेमी

  4. 30 cm/सेमी
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

For transverse common tangent of length 7 between circles with radii 11 and 13, the distance between centers d satisfies: d² = (r1 + r2)² + (transverse tangent)² = (11 + 13)² + 7² = 24² + 7² = 576 + 49 = 625, so d = 25. Option B is correct.

Multiple choice

In a circle of radius 3 units, a diameter AB intersects a chord of Length 3 units. perpendicularly at P. If AP > BP, then AP is equal to approximately. 3 इकाई त्रिज्या के एक वृत्त में व्यास AB एक 3 इकाई की जीवा को लम्बवत् बिन्दु P पर काटता है। यदि AP>BP तो AP का लगभग मान है-

  1. 4.5

  2. 5.6

  3. 2.7

  4. 4.2
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let O be center, AB is diameter (length 6). Let chord CD = 3, perpendicular to AB at P. If AP > BP, then AP > 3. By chord theorem: CP × PD = AP × BP. Since P is midpoint of chord (perpendicular from center bisects chord), CP = PD = 3/2 = 1.5. So 1.5² = AP × BP = AP × (6-AP). 2.25 = 6AP - AP². AP² - 6AP + 2.25 = 0. AP = (6 ± √(36-9))/2 = (6 ± √27)/2 = (6 ± 5.196)/2. AP = 5.598 or AP = 0.402. Since AP > BP, AP > 3, so AP ≈ 5.6, matching option B.

Multiple choice

Two chords AB, CD of a circle with centre O interest each other at P. ∠ADP = 230 and ∠APC = 700, then the ∠BCD is- O केन्द्र वाले एक वृत्त की दो जीवाएँ AB, CD एक-दूसरे का P पर प्रतिच्छेद करती है। ∠ADP = 230 और ∠APC = 700, तो ∠BCD है-

  1. 450

  2. 470

  3. 570

  4. 670
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

In intersecting chords, angles subtended by the same arc are equal. ∠ADP = ∠ABD = 23° (angles in same segment). ∠APC and ∠BPC are vertically opposite, so ∠BPC = 70°. In triangle BPC, ∠PBC = 180° - 70° - 23° = 87°. ∠BCD = ∠BCP = ∠PBC (both subtended by chord PC) = 47°. This uses cyclic quadrilateral properties and angle sum.