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

A chord of length 24 cm is at a distance of 5 cm from the center of the circle. The length of the chord of the same circle which is at a distance of 12 cm from the center is- 24 सेमी लम्बाई की एक जीवा वृत्त के केंद्र से 5 सेमी की दूरी पर स्थित है। समान वृत्त के जीवा की लम्बाई ज्ञात कीजिये जोकि वृत्त के केंद्र से 12 सेमी की दूरी पर स्थित है।

  1. 10

  2. 5

  3. 20

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

Using the property that perpendicular distance from center to a chord bisects the chord: half-chord = √(r² - d²). For first chord: 12 = √(r² - 5²), so 144 = r² - 25, giving r² = 169, r = 13 cm. For second chord at 12 cm: half-chord = √(13² - 12²) = √(169 - 144) = √25 = 5. Therefore, full chord length = 2 × 5 = 10 cm.

Multiple choice

AB and AC are two chords of a circle. The tangents at B and C meet at P. If ∠ BAC=540. Then find the measure of ∠ BPC. AB और AC एक वृत्त के दो जीवायें हैं। B और C पर खीची स्पर्श रेखाएं P पर मिलती हैं। यदि ∠ BAC=540 तो फिर ∠ BPC का माप बताइए।

  1. 1440

  2. 360

  3. 1260

  4. 900
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Using the alternate segment theorem: ∠BPC = 180° - ∠BAC. But ∠BAC = 54° (likely notation error - should be 54° not 540°). So ∠BPC = 180° - 54° = 126°. The angle between the tangents equals the supplement of the angle between the chords. In quadrilateral ABPC, opposite angles sum to 180°.

Multiple choice

The perimeters of two similar triangles are 30 cm. and 20 cm. respectively. If one side of the first traiangle is 12 cm., then find the corresponding side of the second triangle. दो समरूप त्रिभुजों का परिमाप क्रमशः 30 सेमी. और 20 सेमी. है। यदि पहले त्रिभुज की एक भुजा 12 सेमी.है तो दूसरे त्रिभुज की संगत भुजा ज्ञात कीजिए।

  1. 8 cm./सेमी.

  2. 6 cm./सेमी.

  3. 10 cm./सेमी.

  4. 9 cm./सेमी.
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

For similar triangles, ratio of perimeters = ratio of corresponding sides. Perimeter ratio = 30:20 = 3:2. So corresponding side ratio is 3:2. If first triangle's side is 12 cm, second's corresponding side = 12×2/3 = 8 cm. This uses the fundamental property that similar triangles have proportional corresponding dimensions.

Multiple choice

The area of two similar triangle respectively 9 cm2. and 16 cm2. Find the ratio of their corresponding sides. दो समरूप त्रिभुजों के क्षेत्रफल क्रमशः 9 सेमी2. और 16 सेमी2. है, तो उनके संगत भुजाओं का अनुपात क्या है?

  1. 3:4

  2. 4:3

  3. 2:3

  4. 4:5
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

For similar triangles, the ratio of their areas equals the square of the ratio of their corresponding sides. If area ratio is 9:16, then side ratio = √9:√16 = 3:4. This is because area scales with the square of linear dimensions - if sides are in ratio k, areas are in ratio k². This property applies to all similar 2D figures, not just triangles. Option C (2:3) would give area ratio 4:9, and D (4:5) would give 16:25.

Multiple choice

What will be the distance between two parallel chords of length 8 cm. each in a circle of diameter 10 cm.? 10 सेमी. व्यास वाले एक वृत्त में प्रत्येक 8 सेमी. लंबी, दो समानांतर जीवाओं के बीच की दूरी कितनी होगी?

  1. 6 cm./सेमी

  2. 7 cm./सेमी

  3. 8 cm./सेमी

  4. 5.5 cm./सेमी
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

For a circle of radius 5 cm, chord length 8 cm gives perpendicular distance from center = √(5² - 4²) = 3 cm (half the chord is 4 cm). Both chords are parallel and equidistant from center, so distance between chords = 3 + 3 = 6 cm.

Multiple choice

The length of the direct common tangent of two circles of radii 2 cm and 8 cm is 8 cm. Then the distance between the centers of the circles is

  1. 8 cm

  2. 10 cm

  3. 6 cm

  4. 12 cm
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

For two circles with radii r = 2 cm and R = 8 cm, the direct common tangent length L = 8 cm. Using the formula L = √(d² - (R - r)²) where d is the distance between centers: 8 = √(d² - (8 - 2)²) = √(d² - 36). Squaring: 64 = d² - 36, so d² = 100, and d = 10 cm. Option B is correct. Options A (8 cm), C (6 cm), and D (12 cm) are incorrect distances.

Multiple choice

A circle is inscribed in a quadrilateral ABCD. If the lengths of AB, BC, CD are 7 cm, 8.5 cm and 9.2 cm respectively, then the length (in cm) of DA is एक चतुर्भुज ABCD में कोई वृत्त अंतर्निहित है। यदि AB, BC और CD की लंबाई क्रमशः 7 सेमी, 8.5 सेमी तथा 9.2 सेमी है, तो DA की लंबाई (सेमी में) कितनी है

  1. 10.7 cm./सेमी.

  2. 16.2 cm./सेमी.

  3. 7.7 cm./सेमी.

  4. 7.2 cm./सेमी.
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

In a tangential quadrilateral, sum of opposite sides are equal: AB + CD = BC + DA. So 7 + 9.2 = 8.5 + DA, giving DA = 7.7 cm.

Multiple choice

In a circle of radius of 3 units, a diameter AB intersects a chord of length 2 units perpendicular at P. If AP > BP, then what is the ratio of AP to BP? 3 इकाई त्रिज्या वाले एक वृत्त में, एक व्यास AB, 2 मीटर लम्बाई की एक जीवा को लम्बवत बिंदु P पर प्रतिच्छेद करता है । यदि AP > BP है, तो AP का BP से क्या अनुपात है?

  1. 3+ 8 :3− 8

  2. 3+ 10 :3− 10

  3. 3+ 3 :3− 3

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

In a circle, when a diameter intersects a chord perpendicularly at P: AP × BP = (chord half-length)² = 1² = 1. Also AP + BP = 2r = 6. Solving: If AP = x, BP = 6-x, then x(6-x) = 1, giving x² - 6x + 1 = 0. Solving: x = (6 ± √(36-4))/2 = (6 ± √32)/2 = 3 ± 2√2 = 3 ± √8. Since AP > BP, AP = 3 + √8 and BP = 3 - √8, giving ratio (3+√8):(3-√8).

Multiple choice

If the area of isosceles triangle ABC is 108 cm.2 and distance of base BC from centroid is 6 cm. and length of equal sides are 6 10 cm. find the length of base. यदि समद्विबाहु त्रिभुज ABC का क्षेत्रफल 108 सेमी.2 है और आधार BC केन्द्रक से 6 सेमी. तथा बराबर भुजाओ की लंबाई 6 10 सेमी. है। तो आधार की लंबाई क्या होगी?

  1. 6 cm./सेमी.

  2. 12 cm./सेमी.

  3. 18 cm./सेमी.

  4. 24 cm./सेमी.
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Area of triangle ABC = 108 cm². Distance of base BC from centroid G = 6 cm. The centroid divides the median in ratio 2:1, so the height h from A to BC satisfies h/3 = 6, giving h = 18 cm. Area = (1/2) × base × height = (1/2) × b × 18 = 108, so base b = 108 × 2/18 = 12 cm. The equal sides' information is consistent but not needed for this calculation.

Multiple choice

If the chord of an arc of a circle is of length m, the height of the arc is n and the radius of the circle is r. Then, which one of the following is correct? यदि एक वृत्त की चाप की जीवा की लंबाई m मीटर की है, और चाप की ऊंचाई n है और वृत्त की त्रिज्या r है । तो, निम्नलिखित में से कौन सा सही है?

  1. n (2r - n) = m2

  2. n (2r - n) = 4m2

  3. 2n (2r - n) = m2

  4. 4n (2r - n) = m2
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The relationship between chord length m, sagitta (height) n, and radius r in a circle is m² = 4n(2r - n). This formula comes from the fact that the radius to the chord's midpoint is perpendicular to the chord, forming a right triangle with half the chord (m/2) and the distance from center to chord (r-n). By the Pythagorean theorem: (m/2)² + (r-n)² = r², which simplifies to m² = 4n(2r - n). Option A is incorrect as it's missing the factor of 4, and option C is incorrect as it has 2n instead of 4n.

Multiple choice

What is the area of the right-angled triangle? Statement I: The perimeter of the triangle is 5 times the base. Statement II: One of the angles of the triangle is 60°. Statement III: The length of the hypotenuse is 4 cm.

  1. Only II and III

  2. All statements together are sufficient

  3. None of these

  4. I and II or III

  5. All statements together are not sufficient
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Statement II gives one angle as 60°, which in a right triangle means the other acute angle is 30°. Statement III gives the hypotenuse as 4 cm. In a 30-60-90 triangle, if hypotenuse = 4 cm, then shorter side = 2 cm and longer side = 2√3 cm. Area = (1/2) × 2 × 2√3 = 2√3 cm². Statement I about perimeter being 5 times the base is unnecessary.

Multiple choice

SR is a direct common tangent to the circles of radii 8 cm and 3 cm respectively, their centres being 13 cm apart. If the points S and R are the respective points of intersect, then the length of SR is SR क्रमशः 8 सेमी और 3 सेमी त्रिज्या वाले वृत्तों एक एक सीधी उभयनिष्ठ स्पर्श रेखा है, उनके केंद्र एक दूसरे से 13 सेमी दूरी पर हैं । यदि बिन्दु S और R क्रमशः प्रतिछेद बिन्दु हैं, तो SR की लम्बाई है

  1. 12 cm./सेमी.

  2. 11 cm./सेमी.

  3. 17 cm./सेमी.

  4. 10 cm./सेमी.
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

For direct common tangent to circles with radii r₁=8, r₂=3 and center distance d=13, use SR² = d² - (r₁-r₂)². SR² = 13² - (8-3)² = 169 - 25 = 144. Therefore SR = √144 = 12 cm.

Multiple choice

N is the foot of the perpendicular from a point P of a circle with radius 7 cm, on a diameter AB of the circle. If the length of the chord PB is 12 cm, the distance of the point N from the point P is एक वृत्त जिसकी त्रिज्या 7 सेमी है उसके बिन्दु P से व्यास AB पर एक लम्ब डाला गया है जिसका आधार N है यदि जीवा PB की लम्बाई 12 सेमी. है तो बिन्दु N से बिन्दु P की दूरी है

  1. 6 5 7

  2. 12 2 7

  3. 3 5 7

  4. 10 2 7
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

In a circle of radius 7, chord PB = 12. Distance from center O to chord: d = √(7² - 6²) = √13. Coordinate approach: Place circle at origin, AB as x-axis. If P = (x,y) on circle and PB is chord to B = (7,0), with PB = 12, we find PN (distance from P to diameter AB). Using circle geometry: PN = (OP × PB)/radius = (7 × 12)/7 = 72/7 = 10 2/7.

Multiple choice

OC is perpendicular to the chord AB of a circle with center O.OC is produced to meet the circle in D.If AB = 8 cm. and CD = 2 cm., then the radius of the circle is -

  1. 4 cm.

  2. 6 cm.

  3. 5 cm.

  4. 8 cm.
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

OC is perpendicular to chord AB, so C bisects AB. AC = 4 cm. Let radius = r, then OC = r - CD = r - 2. In right triangle OAC: OA² = OC² + AC², so r² = (r - 2)² + 4² = r² - 4r + 4 + 16. Cancelling r²: 0 = -4r + 20, giving r = 5 cm. This matches option C.

Multiple choice

AB is a chord of a circle whose centre is O. P is the point on the circle such that OP ⊥ AB and OP intersects AB at the point in. If AB = 8 cm and MP = 2 cm, then the radius of the circle is AB एक वृत्त की एक जीवा है जिसका केंद्र O है । वृत्त पर कोई बिंदु P ऐसे स्थित है कि OP ⊥ AB और OP, AB को बिंदु पर प्रतिच्छेद करता है। यदि AB = 8 सेमी और MP = 2 सेमी, तो वृत्त की त्रिज्या है -

  1. 10 cm./सेमी.

  2. 6 cm./सेमी.

  3. 5 cm./सेमी.

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

Since OP ⟂ AB, OP bisects AB at M. So AM = MB = 4 cm. In right triangle OMP: OM² + MP² = OP². Also OM² + AM² = OA² (radius). Given AB = 8 cm, so AM = 4 cm. MP = 2 cm. OM² = OA² - 16. Also OM² + 4 = (OA - 2)². Solving: OA² - 16 + 4 = OA² - 4OA + 4, so OA² - 12 = OA² - 4OA + 4, therefore 4OA = 16 and OA = 5 cm. Options A, B, D are calculation errors.