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 chord of a circle is equal to its radius. The angle subtended by this chord at the minor arc of the circle is : एक वृत्त की जीवा उसकी त्रिज्या के बराबर है। वृत्त के लघु चाप पर इस जीवा द्वारा अंतरित कोण कितना होगा?

  1. 60o

  2. 120o

  3. 150o

  4. 135o
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

When chord = radius r, the triangle formed by two radii and the chord is equilateral (all sides = r). Central angle = 60°. The angle subtended at the minor arc = 180° - 60° = 150° (opposite angles of cyclic quadrilateral sum to 180°).

Multiple choice

A,B and C are three points on a circle.The tangent at C meets BA produced at T. Given ∠ ATC = 360 and ∠ ACT = 480, the angle subtended by AB at the centre of the circle is - A,B तथा C एक वृत्त पर तीन बिंदु है। उसमें C पर खींची स्पर्श रेखा BA को बढ़ाकर T पर मिल जाती है। तद्नुसार, यदि ∠ ATC = 360 तथा ∠ACT = 480 दिया हो,तो AB द्वारा वृत्त के केंद्र पर बनाया कोण कितना हो जाएगा?

  1. 48o

  2. 36o

  3. 96o

  4. 120o
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

By the alternate segment theorem, ∠ABC = ∠ACT = 48°. In triangle ABC, ∠BAC = 180° - ∠ATC - ∠ACT = 180° - 36° - 48° = 96°. The angle subtended by chord AB at the center is twice the angle subtended at any point on the circumference, so central angle = 2 × ∠ACB. ∠ACB = 180° - 96° - 48° = 36°. Central angle = 2 × 48° = 96°.

Multiple choice

D and E are the mid points of two cords AB and AC of a circle with a centre O. The line OD and OE are produced to meet the circle respectively at the point F and G. H is a point on the major arc between F and G of circle. If ∠BAC = 40o , then ∠ FHG=? O केंद्र वाले वृत्त की दो जीवाएं AB और AC के मध्य बिंदु D और E है |रेखाएं OD और OE आगे बढ़ते हुए वृत्त के बिंदु F और G पर मिलती है | वृत्त के वृहद् चाप F और G पर एक बिंदु H है | यदि ∠BAC = 40o , तब ∠ FHG=?

  1. 75o

  2. 70o

  3. 65o

  4. 80o
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

D is the midpoint of chord AB, so OD is perpendicular to AB (property: line from center to midpoint of chord is perpendicular to chord). Similarly, OE is perpendicular to AC. This means ∠BAC = 40° is an inscribed angle subtending arc BC. Central angle ∠BOC = 2 × ∠BAC = 80°. Points F and G are on the circle, and H is on the major arc FG. Using cyclic quadrilateral properties, ∠FHG = (180° - ∠FOG)/2. After calculation: ∠FHG = 180° - ∠F O G/2 = 180° - 40° = 140°, then using chord properties: ∠FHG = 70°.

Multiple choice

? ABC is a right - angled triangle at A with AB=6cm. and AC= 8cm. A circle with centre has been inscribed inside the triangle. Then find the radius of the circle.

  1. 3cm

  2. 2.5cm

  3. 2cm

  4. 1cm

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

Triangle with sides 6, 8 is a 3-4-5 triangle scaled by factor 2. Hypotenuse = 10cm. For a right triangle, inradius r = (a + b - c)/2 where c is hypotenuse. So r = (6 + 8 - 10)/2 = 4/2 = 2cm. Alternatively, area = 24 = r × s where s = semiperimeter = 12, giving r = 2. Options A and D overestimate the inradius.

Multiple choice

AB is a chord of the circle with centre O, such that OP is perpendicular to AB and OP intersect AB at point M. If AB = 8cm, MP = 2cm then find the radius of the circle O केंद्र वाले वृत्त की जीवा AB इस प्रकार है कि OP, AB पर लम्ब है और OP ,AB को बिंदु M पर प्रतिच्छेदित करता है .यदि AB=8 सेमी ,MP=2 सेमी तो वृत्त की त्रिज्या ज्ञात कीजिए |

  1. 7 cm / सेमी

  2. 5 cm / सेमी

  3. 4 cm / सेमी

  4. 6 cm / सेमी
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

In the circle, OM is perpendicular to chord AB. M is midpoint of AB, so AM = MB = 4 cm. Let OM = d. In right triangle OMA: OA² = OM² + AM² = d² + 16 = r². Given MP = 2 cm. Since OM + MP = OP = r (if P lies on radius extended), we have d + 2 = r. Substituting: r² = (r-2)² + 16 = r² - 4r + 4 + 16, giving 4r = 20, so r = 5 cm.

Multiple choice

If O is the centre of a circle and AB is a chord of the circle at a distance half of the radius of circle from the centre O.Find the value of the ∠BAO. यदि एक वृत्त का केंद्र O है तथा इसमें एक जीवा AB हैं जिसकी केंद्र O से दूरी वृत्त की त्रिज्या से आधी हैं। ∠BAO का मान ज्ञात कीजिए।

  1. 60o

  2. 30o

  3. 45o

  4. 90o
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let M be the midpoint of chord AB. In right triangle OMA: OM = r/2 (given, half the radius), OA = r (radius). Using trigonometry: cos(∠AOM) = OM/OA = (r/2)/r = 1/2, so ∠AOM = 60°. Since OM is perpendicular to AB, ∠BAO = 90° - 60° = 30°. This uses the property that the line from the center to a chord's midpoint is perpendicular to the chord.

Multiple choice

In a triangle ABC, AD is the bisector of ∠BAC. If AB = 2.8 cm, AC = 4.9 cm and CD = 2.1 cm. then what is the length of BD? त्रिभुज ABC में ∠BAC का समद्विभाजक AD है। यदि AB = 2.8 सेमी. AC = 4.9 सेमी. और CD = 2.1 सेमी. तो BD की लम्बाई क्या है?

  1. 3.7 cm./सेमी.

  2. 0.8 cm./सेमी.

  3. 1.2 cm./सेमी.

  4. 1.0 cm./सेमी.
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

By the Angle Bisector Theorem, in triangle ABC with angle bisector AD, we have: BD/DC = AB/AC. Given AB = 2.8 cm, AC = 4.9 cm, and DC = 2.1 cm. So BD/2.1 = 2.8/4.9, which gives BD/2.1 = 4/7, therefore BD = 2.1 x 4/7 = 1.2 cm.

Multiple choice

Two chords PQ and RS of circle whose center is O, Meets at point T and ∠POR = 60o,∠QOS = 50o, then the value ∠QTS is- केन्द्र "O" वाले वृत्त की दो जीवाएँ PQ एंव RS एक दूसरे को T पर मिलती है , एवं ∠POR = 60o,∠QOS = 50o तो ∠QTS का मान है -

  1. 55o

  2. 45o

  3. 75o

  4. 90o
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

For intersecting chords in a circle, the angle formed at the intersection point equals half the sum of the measures of the arcs intercepted by the angle and its vertical angle. Here, ∠QTS = 1/2(arc(QS) + arc(PR)). Given ∠QOS = 50° and ∠POR = 60°, the sum of these central angles is 110°, and ∠QTS = 110°/2 = 55°.

Multiple choice

AB and CD are two parallel chords on the opposite sides of the circle, if AB = 10 cm CD =24 cm and the radius of the circle is 13 cm. The distance between the chords is? एक वृत्त के केन्द्र के विपरीत जीवायें AB और CD है। यदि AB =10 सेमी. CD = 24 सेमी. और वृत्त की त्रिज्या 13 सेमी. है तो जीवाओं के बीच दूरी क्या होगी ?

  1. 14 cm

  2. 15 cm

  3. 16 cm

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

For chord AB = 10 cm in a circle of radius 13 cm: distance from center d1 = √(13² - 5²) = √(169 - 25) = √144 = 12 cm. For chord CD = 24 cm: distance from center d2 = √(13² - 12²) = √(169 - 144) = √25 = 5 cm. Since chords are on opposite sides of center, total distance = d1 + d2 = 12 + 5 = 17 cm.

Multiple choice

Two tangents TP and TQ are drawn to a circle with centre O from an external point T. Then ∠PTQ equals to - O केंद्र वाले वृत्त पर एक बाह्य बिंदु T से दो स्पर्श रेखा TP और TQ खींची जाती है, तो ∠PTQ बराबर है -

  1. ∠OPQ

  2. 2∠ OPQ

  3. 1/2 ∠ OPQ

  4. 1/3∠ OPQ
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Since TP and TQ are tangents from point T, they are equal in length and OT bisects ∠PTQ. In right triangle OPQ, ∠OPQ is the angle between OP and PQ. The angle ∠PTQ is external to triangle OPQ and equals 2∠OPQ. This is a standard property: the angle between two tangents equals the difference in the arcs they intercept, which is twice the angle in the alternate segment.

Multiple choice

If ΔABC and ΔDEF are similar triangle and AB = 1.2 cm and DE = 1.4 cm. Find the ratio of areas of ΔABC and ΔDEF. यदि ΔABC और ΔDEF समरूप त्रिभुज है और AB = 1.2 सेमी. DE = 1.4 सेमी. है तो ΔABC और ΔDEF के क्षेत्रफलों का अनुपात ज्ञात कीजिये।

  1. 6 : 7

  2. 12 : 11

  3. 9 : 16

  4. 36 : 49
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

For similar triangles, the ratio of areas equals the square of the ratio of corresponding sides. AB/DE = 1.2/1.4 = 6/7. Therefore, Area(ABC)/Area(DEF) = (6/7)² = 36/49.

Multiple choice

The distance between the centres of two circles having radii 4.5cm. and 3.5cm. Respectively is 10cm. What is the length of the transverse common tangent length of these circles? 4.5 सेमी. और 3.5 सेमी. वाली त्रिज्या वाले दो वृत्तों के केन्द्र के बीच की दूरी 10 सेमी. है। इन वृत्तों की उभयनिष्ठ प्रतिच्छेदी स्पर्श रेखा की लम्बाई क्या है?

  1. 8 cm/सेमी

  2. 7 cm/सेमी

  3. 6 cm/सेमी

  4. 12 cm/सेमी
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

For two circles with radii r1 and r2 and distance d between centers, the transverse common tangent length = √(d² - (r1 + r2)²) = √(10² - (4.5 + 3.5)²) = √(100 - 64) = √36 = 6 cm.

Multiple choice

The length of the common chord of two circles of radii 30cm. and 40cm. whose centres are 50 cm. apart, in (in cm.) त्रिज्या 30 सेमी. और 40 सेमी. वाले दो वृत्तों की उभयनिष्ठ जीवा की लम्बाई, जिनके केन्द्र 50 सेमी. दूरी पर हैं, (सेमी. में)

  1. 12

  2. 24

  3. 36

  4. 48
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

For two intersecting circles with radii r1=30, r2=40 and distance d=50, the common chord length = 2 × √(r1² - (d² + r1² - r2²)²/(4d²)). Using this formula or noting that 30-40-50 is a right triangle, the perpendicular distance from centers to chord gives chord length = 48 cm.

Multiple choice

One chord of a circle is known to be 14 cm. what is the Area of a circle if length of perpendicular to chord from the centre of circle is 5 cm. एक वृत्त की जीवा की लम्बाई 14 सेमी. है और उसके केन्द्र और जीवा की लम्बवत दूरी 5 सेमी. है तो वृत्त का क्षेत्रफल क्या होगा ?

  1. 84π

  2. 74π

  3. 75π

  4. 85π
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Using chord property: The perpendicular from the center to a chord bisects it. Half-chord = 7 cm. In right triangle: r² = 5² + 7² = 25 + 49 = 74. Area = πr² = π × 74 = 74π. (Note: The radius is √74 cm, approximately 8.6 cm, which gives area 74π sq.cm ≈ 232.5 sq.cm).

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 portion is- 8 सेमी. भुजा वाले चारों कोने से 4 सेमी. त्रिज्या वाले वृ़त्त के चाप के आकार में चार टुकड़े काटकर अलग कर दिए हैं, तदनुसार शेष आकृति क क्षेत्रफल कितना है ?

  1. (4–π) cm2/सेमी2

  2. 16(4–π) cm2/सेमी2

  3. 8(4–π) cm2/सेमी2

  4. 7(4–π) cm2/सेमी2
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Original square area = 8² = 64 cm². Four quarter-circles (or one full circle) of radius 4 cm are removed from corners. Area removed = π × 4² = 16π cm². Remaining area = 64 - 16π = 16(4 - π) cm². The key insight is recognizing that four quarter-circles equal one complete circle.