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

$ABC$ is a triangle inscribed in a circle with centre $O$. Let $\alpha =\angle BAC$, where ${ 45 }^{ o }<\alpha <{ 90 }^{ o }$. Let $ \beta =\angle BOC$. Which one of the following is correct?
  1. $\cos { \beta } =\cfrac { 1-\tan ^{ 2 }{ \alpha } }{ 1+\tan ^{ 2 }{ \alpha } } $
  2. $\cos { \beta } =\cfrac { 1+\tan ^{ 2 }{ \alpha } }{ 1-\tan ^{ 2 }{ \alpha } } $
  3. $\cos { \beta } =\cfrac { 2\tan { \alpha } }{ 1+\tan ^{ 2 }{ \alpha } } $
  4. $\sin { \beta } =2\sin ^{ 2 }{ \alpha } $
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

In a circle, the angle at the center (beta) is twice the angle at the circumference (alpha) subtended by the same arc, so beta = 2 * alpha. Thus, cos(beta) = cos(2 * alpha). Using the double angle identity, cos(2 * alpha) = (1 - tan^2(alpha)) / (1 + tan^2(alpha)).

Multiple choice

The radius and length of an arc of a circle are 35 cm and 22 cm respectively The angle subtended by the arc at the centre of the circle is
  1. $\displaystyle 72^{\circ} $
  2. $\displaystyle 45^{\circ} $
  3. $\displaystyle 36^{\circ} $
  4. $\displaystyle 40^{\circ} $
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Arc length L = r * theta (in radians). 22 = 35 * theta. theta = 22/35 radians. To convert to degrees: (22/35) * (180/pi) = (22/35) * (180 / (22/7)) = (22/35) * (180 * 7 / 22) = 180 / 5 = 36 degrees.

Multiple choice

In a circle central angle is equal to

  1. inscribed angle

  2. intercepted arc

  3. tangent

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

A central angle in a circle is defined as the angle subtended by an arc at the center of the circle, which is equal to the measure of the intercepted arc.