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

Find the area of a regular pentagon whose each side measure $6\;cm$ and the radius of the inscribed circle is $4\;cm$.
  1. $60\;cm^2$
  2. $80\;cm^2$
  3. $60\;m^2$

  4. None of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The area of a regular polygon is given by the formula Area = (1/2) * perimeter * apothem. The perimeter of a pentagon with side 6 cm is 5 * 6 = 30 cm, and the apothem (radius of the inscribed circle) is 4 cm. Thus, Area = (1/2) * 30 * 4 = 60 cm^2.

Multiple choice

Two parallel chords $AB$ and $CD$ are drawn on the same side of centre $O$ of a circle. Radius of circle is $65: m$, length of chords are $112: m$ and $126: m$. Find the area of $\square ABCD$.
  1. $2023\:m^2$
  2. $2123\:m^2$
  3. $2223\:m^2$
  4. $2323\:m^2$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The distance of the chords from the center are found using the Pythagorean theorem: d1 = sqrt(65^2 - 56^2) = 33 and d2 = sqrt(65^2 - 63^2) = 16. Since they are on the same side, the distance between them is 33 - 16 = 17. The area of the trapezoid ABCD is ((AB + CD) / 2) * height = ((112 + 126) / 2) * 17 = 119 * 17 = 2023.

Multiple choice

A rectangle ABCD is inscribed in a circle. Let PQ be the diameter of the circle parallel the side AB. If $\angle BPC$ = $30^\circ $ , then the ratio of the area of rectangle to the area of circle is
  1. ${{\sqrt 3 } \over \pi }$
  2. ${{\sqrt 3 } \over {2\pi }}$
  3. ${3 \over \pi }$
  4. ${{\sqrt 3 } \over {9\pi }}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let radius be R. Rectangle area = 2R^2 * sin(theta) * cos(theta) or similar geometry. Given the angle, the ratio simplifies to sqrt(3)/pi.

Multiple choice

If the chords of contact of points on ${x}^{2}+{y}^{2}={a}^{2}$ with respect to the circle ${x}^{2}+{y}^{2}={b}^{2}$ touch the circle ${x}^{2}+{y}^{2}={c}^{2}$, then ${a},\ {b},\ {c}$ are in

  1. A.P.

  2. G.P.

  3. H.P.

  4. A.G.P.
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The chord of contact of a point on x^2 + y^2 = a^2 with respect to x^2 + y^2 = b^2 is xx' + yy' = b^2. The condition for this line to touch x^2 + y^2 = c^2 is that the distance from the origin is c. This leads to the relationship b^2 = ac, meaning a, b, c are in G.P.

Multiple choice

The length of the common chord of the ellipse $\cfrac { { (x-1) }^{ 2 } }{ 9 } +\cfrac { { (y-2) }^{ 2 } }{ 4 } =1$ and the circle $ { (x-1) }^{ 2 }+{ (y-2) }^{ 2 }=1$ is:
  1. $1$
  2. $3$
  3. $8$

  4. None of these
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The ellipse is ((x-1)^2)/9 + ((y-2)^2)/4 = 1 and the circle is (x-1)^2 + (y-2)^2 = 1. Since the circle is entirely inside the ellipse (except at points where the ellipse is smaller, but here the ellipse is larger in both axes), they do not intersect. The common chord length is undefined or 0.

Multiple choice

The radical centre of the circles drawn on the focal chords of the parabola $y^{2}=4ax$ as diameters is
  1. $(-a, 0)$
  2. $(a, 0)$
  3. $(0, 0)$
  4. $(a, a)$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

For a parabola y^2 = 4ax, a focal chord passing through (a, 0) with slope m has endpoints (at^2, 2at) and (a/t^2, -2a/t). A circle with this chord as diameter has the equation (x - at^2)(x - a/t^2) + (y - 2at)(y + 2a/t) = 0. The radical center of such circles for different focal chords is the origin (0, 0).