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

If radii of two concentric circles are 4 cm and 5 cm, then the length of each chord of one circle which is tangent to the other circle is

  1. 3 cm

  2. 6 cm

  3. 9 cm

  4. 1 cm
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The chord of the larger circle tangent to the smaller circle forms a right triangle with the radius of the smaller circle (4 cm) and the radius of the larger circle (5 cm). By the Pythagorean theorem, half the chord length is sqrt(5^2 - 4^2) = sqrt(25 - 16) = sqrt(9) = 3 cm. The full chord length is 2 * 3 = 6 cm.

Multiple choice

If two tangents inclined at an angle $60^0$ are drawn to a circle of radius 3 cm. The distance of the chord joining the point of contact of the tangents to the circle is $ 1.5cm. $ Then length of each tangent is equal to
  1. $\frac{3}{2}\sqrt{3}$ cm

  2. 6 cm

  3. 3 cm
  4. $3\sqrt{3}$ cm
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Let the tangents be PA and PB. In the triangle formed by the center O, the point of contact A, and the external point P, we have a right triangle OAP. Angle OPA = 30 degrees (half of 60). tan(30) = OA / AP = 3 / AP. AP = 3 / tan(30) = 3 / (1/sqrt(3)) = 3 * sqrt(3).

Multiple choice

Draw two concentric circles of radii 3 cm and 5 cm. Taking a point on outer circle construct the pair of tangents to the other. Measure the length of a tangent and verify it by actual calculation.

  1. 3 cm

  2. 4 cm

  3. 5 cm

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

The radius of the inner circle is 3 cm and the outer is 5 cm. A tangent from the outer circle to the inner circle forms a right triangle with the radius of the inner circle (3 cm) and the radius of the outer circle (5 cm) as the hypotenuse. Tangent length = sqrt(5^2 - 3^2) = sqrt(25 - 9) = sqrt(16) = 4 cm.

Multiple choice

The points A, B, and C are on a circle O. The tangent line at A and the secant BC intersect at P, B lying between C and P. If $\overline{BC}$ = 20 and $\overline{PA}$ = $10\sqrt{3}$ ,then $\overline{PB}$ equals:
  1. $5$
  2. $10$
  3. $10\sqrt{3}$
  4. $20$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

By the Tangent-Secant Theorem, PA^2 = PB * PC. Let PB = x. Then PC = x + 20. (10*sqrt(3))^2 = x(x+20). 300 = x^2 + 20x. x^2 + 20x - 300 = 0. (x+30)(x-10) = 0. Since length must be positive, x = 10.