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 length of the common chord of two circles of radii 15 cm. and 20 cm. whose centres are 25 cm. apart is? (in cm.) 15 सेमी. और 20 सेमी. त्रिज्या वाले दो वृत्तों की एक उभयनिष्ठ जीवा की लम्बाई (सेमी. में) क्या है जिनके केन्द्रों की दूरी 25 सेमी. है?

  1. 15 cm./सेमी.

  2. 20 cm./सेमी.

  3. 24 cm./सेमी.

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

For two circles with radii 15 cm and 20 cm, and center distance 25 cm, note that 15² + 20² = 225 + 400 = 625 = 25², so the circles intersect at right angles. The common chord length is 2√(15² - d₁²), where d₁ = 15²/25 = 9. So chord length = 2√(225 - 81) = 2√144 = 2 × 12 = 24 cm.

Multiple choice

The radius of two concentric circles are 6 cm and 10 cm. If the chord of the greater circles be a tangent to the smaller circle, then the length of that chord is - दो संकेद्रीय वृत्तों की त्रिज्यायें 6 सेमी और 10 सेमी है। यदि बड़े वृत्त की जीवा, छोटे वृत्त की स्पर्श रेखा है तो उस जीवा की लम्बाई है -

  1. 12 cm./सेमी.

  2. 13 cm./सेमी.

  3. 16 cm./सेमी.

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

The perpendicular from the center to the chord bisects it and passes through the point of tangency. Using Pythagoras theorem: half the chord length = sqrt(radius of larger circle^2 - radius of smaller circle^2) = sqrt(100 - 36) = sqrt(64) = 8 cm. Therefore, the full chord length is 16 cm.

Multiple choice

What is the circumference of circle inscribed in a triangle having length of sides 35 cm, 44 cm and 75 cm? उस वृत्त की परिधि क्या है, जो 35 सेमी., 44 सेमी. और 75 सेमी. त्रिभुज में खींचा जाता है?

  1. 8 π cm./सेमी.

  2. 12 π cm./सेमी.

  3. 16 π cm./सेमी.

  4. 7.4 π cm./सेमी.
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

For sides 35, 44, 75: s = (35+44+75)/2 = 77. Area = sqrt(77*42*33*2) = sqrt(212052) ≈ 460.5 cm². Inradius r = Area/s = 460.5/77 ≈ 5.98. Circumference = 2πr = 2π*4 = 8π. Wait, let me recalculate: sqrt(77*42*33*2) = sqrt(212052) ≈ 460.5. But notice: 35+44 = 79, which is > 75, so triangle is valid. Actually, r = A/s. But let's use formula: r = (a+b-c)/2 for any triangle? No, that's for right triangles. For any triangle, r = A/s. Let me recalculate: 77*42*33*2 = 212052. sqrt(212052) = 460.5. r = 460.5/77 = 5.98. Circumference = 2π*5.98 ≈ 37.6 = 12π (since 12π ≈ 37.7). Option B (12π) is correct.

Multiple choice

In a circle of radius 5 cm, a chord is drawn 3 cm. from the centre. If a chord half the length of the original chord were drawn, its distance in centimetres from the centre would be - 5 सेमी. त्रिज्या वाले एक वृत्त में, केन्द्र से 3 सेमी. दूर एक जीवा खींची गयी है। यदि एक जीवा जो मूल जीवा की लम्बाई की आधी है, खींची जाये तो केन्द्र से इसकी दूरी (सेमी. में )होगी-

  1. 13.6857

  2. 13.8675

  3. 31.6875

  4. 13.6875
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

For radius 5, chord at 3 cm: half-chord = 4 cm (3-4-5 triangle). Original chord = 8 cm. Half chord = 4 cm. Distance from center: sqrt(5² - 4²) = 3 cm. This tests circle geometry: chord length and distance from center relationship.

Multiple choice

Two circle C (O, r) and C' (O', r1) intersect each other at P and Q respectively. OO' cut common chord at M such that OQ = 17 cm and OM = 15 cm then find the length of PQ. दो वृत्त C (O, r) और C' (O', r1) एक दूसरे को बिन्दु P और Q पर प्रतिच्छेद करते हैं रेखा OO' उभयनिष्ठ जीवा PQ का M पर इस प्रकार प्रतिच्छेद करती है कि OQ = 17 सेमी और OM = 15 सेमी. तो PQकी माप ज्ञात कीजिए।

  1. 8 cm/ सेमी.

  2. 12 cm/ सेमी.

  3. 20 cm/ सेमी.

  4. 16 cm/ सेमी.
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

OO' is the perpendicular bisector of common chord PQ. Triangle OMQ is right-angled at M. OQ = 17 cm (radius), OM = 15 cm (distance from center to chord). Using Pythagoras: PM² = OQ² - OM² = 17² - 15² = 289 - 225 = 64, so PM = 8 cm. Since M is midpoint of PQ, total PQ = 2 × PM = 16 cm.

Multiple choice

In a circle O is centre diameter PQ cut chord MN at O'. If ∠NOQ=108o and QM=QN then find the value of ∠PO'N. एक वृत्त में O केन्द्र है जीवा MN और व्यास PQ एक दूसरे को O' पर काटते हैं। यदि ∠NOQ=108o और QM=QN है तो ∠PO'N का मान ज्ञात कीजिए।

  1. 72o

  2. 108o

  3. 90o

  4. 36o
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Given ∠NOQ = 108° and QM = QN, triangle NQM is isosceles at Q. In circle geometry, when a diameter (PQ) intersects a chord (MN) at O', if QM = QN, then O' is the midpoint of MN. Since PQ is a diameter passing through center O, and O' lies on MN, the line PO'N is perpendicular to MN. Therefore ∠PO'N = 90°.

Multiple choice

The side of a square is 10 cm. Find the ratio of area of circumscribed and inscribed circles. एक वर्ग की भुजा 10 सेमी हैं। इसके परिवृत्त और अन्तः वृत्त के क्षेत्रफल का अनुपात ज्ञात कीजिए।

  1. 1:2

  2. 2:1

  3. 2:3

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

For square with side 10 cm, circumscribed circle has diameter = diagonal = 10√2 cm, so radius = 5√2 cm. Area = π(5√2)² = 50π. Inscribed circle has diameter = side = 10 cm, so radius = 5 cm. Area = π(5)² = 25π. Ratio of areas = 50π : 25π = 2:1. The circumscribed circle always has twice the area of the inscribed circle for a square.

Multiple choice

There are two chords AB and DC of a circle C (O, r). Chord AB and DC, interest each other at P when produced. If AB = 13 cm. CD=9 cm., PC=6 cm. then find the value of BP. एक वृत्त C (O, r) में दो जीवाएँ AB और DC हैं। जीवा AB और DC एक दूसरे को P पर बढ़ाने पर मिलती हैं। यदि AB =13 सेमी. , CD=9 सेमी. PC=6 सेमी. तो BP का मान ज्ञात कीजिए।

  1. 4 cm./सेमी

  2. 5 cm./सेमी

  3. 6 cm./सेमी

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

For two intersecting chords, the products of the segments are equal: AP × BP = CP × DP. Given AB = 13, so AP = 13 - BP. Also CD = 9 and PC = 6, so PD = 9 - 6 = 3. Then (13 - BP) × BP = 6 × 3 = 18. Solving BP² - 13BP + 18 = 0 gives BP = 4 or 9. Since AB = 13 and BP must be less than AB, BP = 4 cm.

Multiple choice

Two parallel chords of a circle whose diameter is 13 m. are respectively 5m and 12 m in length If both the chords are on the same side of the diameter, then the distance between these chords is- एक वृत्त जिसका व्यास 13 सेमी. है की दो समान्तर जीवाओं की लम्बाई क्रमशः 5 सेमी. और 12 सेमी. है। यदि दोनों जीवाएं व्यास के एक ही तरफ हों, तो इन जीवाओं के बीच की दूरी है-

  1. 5.5

  2. 5

  3. 3.5

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

Radius r = 6.5m. Distance from center to chord with length L is d = √(r²-(L/2)²). For L=5: d₁ = √(42.25-6.25) = 6m. For L=12: d₂ = √(42.25-36) = 2.5m. Both chords same side, so gap = 6-2.5 = 3.5m. Option D (3) forgets 0.5, B (5) is miscalculation, A (5.5) adds instead of subtracts.

Multiple choice

If a chord of a circle of the radius 5 cm. is a tangent to a circle of radius 3 cm., both the circle being concentric, then the length of the chord is - 5 सेमी. त्रिज्या वाले एक वृत्त की जीवा, 3 सेमी. त्रिज्या वाले एक अन्य वृत्त की स्पर्श रेखा है। तद्नुसार यदि दोनों वृत्त संकेन्द्रीय हो तो उस जीवा की लम्बाई है -

  1. 5.6

  2. 7.3

  3. 8.0

  4. 9.0
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The chord of the larger circle (radius 5 cm) touches the smaller circle (radius 3 cm) at one point. The perpendicular from the common center to the chord equals the smaller radius = 3 cm. This forms a right triangle with: half-chord (one leg), 3 cm (other leg), and 5 cm (hypotenuse). By Pythagoras: (half-chord)^2 + 3^2 = 5^2 → (half-chord)^2 = 25 - 9 = 16 → half-chord = 4 cm. Full chord length = 2 × 4 = 8 cm.

Multiple choice

In a circle with centre O, ΔPOQ is inscribed where PO is perpendicular to OQ. If the area of the triangle POQ is 72 units, then the area of the circle is - O केन्द्र वाले वृत्त में, ΔPOQ अन्तर्निहित है जिसमें PO, OQ के लम्बवत् है। यदि POQ का क्षेत्रफल 72 मात्रक है, तो वृत्त का क्षेत्रफल है-

  1. 36π

  2. 72π

  3. 144π

  4. 57π
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

POQ is right-angled at O, with PO and OQ as radii (both equal to r). Area of ΔPOQ = (1/2) × PO × OQ = (1/2)r² = 72. Therefore r² = 144. Area of circle = πr² = 144π. Option C is correct.

Multiple choice

Two chords PQ and RS of a circle with centre O intersect each other at A. If ∠POR = 100o and ∠SPA = 30o then what is the measure of ∠PAR if PR and QS intersect each other on extending at E. O केन्द्र वाले वृत्त में दो जीवायें PQ और RS एक दूसरे को A पर काटती हैं। यदि ∠POR 100o और ∠SPA = 30o तो ∠PAR की माप क्या है? यदि PR और QS एक दूसरे को बढ़ाने पर बिन्दु E पर काटते हैं।

  1. 77o

  2. 80o

  3. 85o

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

Using the intersecting chords theorem, the angle between two chords equals half the sum of the arcs they intercept. Given ∠POR = 100° is the central angle subtended by arc PR, and ∠SPA = 30° at the intersection point A, we can determine that ∠PAR = 80° by applying the vertical angles property and angle sum relationships in the quadrilateral formed by the intersecting chords.

Multiple choice

One chord of a circle is known to be 10.1 cm. The radius of this circle must be. एक वृत्त की एक जीवा 10.1 सेमी. है। इस वृत्त की त्रिज्या होनी चाहिए।

  1. 5 cm/सेमी.

  2. greater than 5 cm/5 सेमी. से बड़ी

  3. greater than or equal to 5 cm/5सेमी. के बराबर या इससे बड़ी

  4. less than 5 सेमी. से छोटी
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The diameter is the longest chord of a circle, so any chord must be less than or equal to the diameter (2 × radius). If chord = 10.1 cm, then 2r ≥ 10.1, meaning r ≥ 5.05 cm. Thus, the radius must be strictly greater than 5 cm (not equal to 5).

Multiple choice

The length of a tangent drawn from a point 10 cm. away from the centre of a circle is 8 cm. Find the perimeter of the circle. एक वृत्त से 10 सेमी. दूर स्थित एक बिन्दु से खींची गयी स्पर्श रेखा की लम्बाई 8 सेमी. है। वृत्त का परिमाप ज्ञात कीजिए।

  1. 4π

  2. 12π

  3. 6π

  4. 128π
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Using the tangent-radius theorem, the tangent is perpendicular to the radius at the point of contact, forming a right triangle. By Pythagoras theorem: radius = sqrt(10^2 - 8^2) = 6 cm. Perimeter = 2πr = 12π.

Multiple choice

PQ is a diameter of the circle and PR is a chord of a circle such that ∠QPR =300. The tangent at R intersects PQ produced in S. Then ∠RQS = ? PQ वृत्त का व्यास है और PR वृत्त पर एक जीवा है ताकि ∠QPR = 300 बिन्दु R पर स्पर्श रेखा PR को बढ़ाने पर बिन्दु S पर प्रतिच्छेद करती है। तो ∠RQS

  1. 600

  2. 900

  3. 300

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

PQ is a diameter, so ∠PRQ = 90° (angle in semicircle). Given ∠QPR = 30°, then ∠PQR = 60°. By alternate segment theorem, ∠PRS = ∠PQR = 60°. In cyclic quadrilateral PQRS, opposite angles sum to 180°, so ∠RQS = 180° - 60° = 120°.