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 circle in which CD has a length of 30 cm, acts as the longest chord of the circle. Points R and S lie on the circle such that the distance from R to D is 18 cm, and the length of segment CR is three times the length of segment CS. Then, the ratio of length of RD to CS is:

  1. 9 : 2

  2. 9 : 4

  3. 9 : 8

  4. 18 : 11
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

CD is diameter (30cm). R, S on circle. RD=18. In right triangle CRD, CR = sqrt(30^2 - 18^2) = 24. CR = 3*CS, so CS = 8. Ratio RD:CS = 18:8 = 9:4.

Multiple choice

A circle is inscribed in an equilateral triangle of side 12 cm. A square is then inscribed inside this circle such that all vertices of the square touch the circle. What is the area of the square?
    1. 16 cm2

  2. 18 cm2

  3. 24 cm2

  4. 36 cm2
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The radius of the inscribed circle of an equilateral triangle with side s is r = s / (2*sqrt(3)). For s=12, r = 12 / (2*sqrt(3)) = 2*sqrt(3). A square inscribed in a circle of radius r has a diagonal d = 2r = 4*sqrt(3). The area of the square is d^2 / 2 = (4*sqrt(3))^2 / 2 = (16*3) / 2 = 24.

Multiple choice

In a circle with center C and radius 6 2 cm, PQ and SR are two parallel chords separated by one of the diameters. If ∠PQC = 45° and the ratio of the perpendicular distance of PQ and SR from C is 3 : 2, then the area, in sq. cm, of the quadrilateral PQRS is

  1. 20(3 + 14 )

  2. 20( 3 2 + 7 )

  3. 4(3 + 14 )

  4. 4(3 2 + 7 )
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Interpreting the garbled radius as 6sqrt(2) cm, the given angle makes the distance of PQ from the center 6 cm, so the other distance is 4 cm. The chord lengths are 12 cm and 4sqrt(14) cm, giving area ((12 + 4sqrt(14))/2) x 10 = 20(3 + sqrt(14)) sq. cm.

Multiple choice

Two circles have radii in the ratio 4 : 2. The length of a direct common tangent is 16 cm, and the distance between their centers is 20 cm. What is the radius of the smaller circle?

  1. 10 cm

  2. 12 cm

  3. 14 cm

  4. 16 cm
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let radii be 4k and 2k. Distance between centers d = 20, tangent L = 16. Formula: L^2 = d^2 - (R - r)^2. 16^2 = 20^2 - (4k - 2k)^2. 256 = 400 - (2k)^2. (2k)^2 = 144. 2k = 12. The smaller radius is 2k = 12.

Multiple choice

From an external point T, two tangents TM and TN are drawn to a circle with center O, touching the circle at M and N, respectively. Points R and S are taken on the segments TM and TN, respectively, such that the line RS is also tangent to the same circle. If angle ROS = 70°, then the measure of angle RTS, in degrees, is ______.

  1. 10

  2. 40

  3. 70

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

Let the circle touch TM at M, TN at N, and RS at P. Then RM=RP and SN=SP. The perimeter of triangle TRS is TM + TN. The angle ROS = 70 degrees implies angle MON = 140 degrees. The angle RTS = 180 - 140 = 40 degrees.