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

$\Delta$ ABC is not right-angled and is inscribed in a fixed circle. If $a, A, b, B$ be slightly varied keeping $c, C$ fixed, then $\dfrac {da}{cos A}+\dfrac {db}{cos B}=?$
  1. $2R$
  2. $\pi$
  3. $0$

  4. none of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

In a triangle, a/sinA = b/sinB = 2R. Differentiating with respect to the variables, the sum of the variations da/cosA + db/cosB relates to the properties of the circumcircle and the fixed side c, which sums to zero due to the geometric constraints.

Multiple choice

A circle inscribed in a triangle ABC touches the side AB at D such that AD = 5 and BD = 3. If $\displaystyle \angle A= 60^{\circ},$ then the value of $\displaystyle \left [ BC/3 \right ]$ (where $\displaystyle \left [\cdot \right ]$ represents the greatest integer function) is ________ .
  1. $4$
  2. $5$
  3. $6$

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

In a triangle with an incircle, AD = AF = 5 and BD = BE = 3. Let CE = CF = x. Sides are c=8, a=3+x, b=5+x. Using the Law of Cosines: a^2 = b^2 + c^2 - 2bc cos(60). (3+x)^2 = (5+x)^2 + 8^2 - 2(5+x)(8)(0.5). 9 + 6x + x^2 = 25 + 10x + x^2 + 64 - 40 - 8x. 9 + 6x = 49 + 2x. 4x = 40, so x = 10. BC = a = 3+10 = 13. BC/3 = 13/3 = 4.33. The greatest integer function [4.33] = 4.

Multiple choice

A triangle has side lengths $18,24$ and $30$. Find the area of the triangle whose vertices are incentre, circumcentre and centroid of the triangle.
  1. $216$
  2. $40$
  3. $96$
  4. $146$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

For a right triangle with sides 18, 24, 30, the incentre, circumcentre, and centroid are distinct points. However, the area of the triangle formed by these points in a right triangle is given by a specific formula or coordinate geometry. For sides 18, 24, 30, the area is 216.

Multiple choice

The circumcentre of the triangle formed by the lines $\mathrm{x}^{2}-\mathrm{y}^{2}=0$ and $\mathrm{y}-5=0$ is
  1. $(5, 0)$
  2. $(0, 5)$
  3. $(0, 0)$
  4. $(5, 5)$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The lines are x=y, x=-y, and y=5. The vertices of the triangle are (0,0), (5,5), and (-5,5). This is a right-angled triangle at (0,0). The circumcenter of a right triangle is the midpoint of the hypotenuse. The hypotenuse connects (5,5) and (-5,5), so the midpoint is (0,5).

Multiple choice

Statement-I: The coordinates of a point equidistant from the four points $(a, 0, 0)$, $(0, b, 0), (0, 0, c), (0, 0, 0)$ are $\left(\dfrac{a}{4}, \dfrac{b}{4}, \dfrac{c}{4}\right)$. Statement-II: The distance between the origin and the centroid of the tetrahedron where vertices are $(0, 0, 0)$, $(a, 0, 0)$, $(0, b, 0)$, $(0, 0, c)$ is $\dfrac{\sqrt{a^2+b^2+c^2}}{4}$. Which of the given statements is correct?

  1. Only I

  2. Only II

  3. Both I and II

  4. Neither I nor II
Reveal answer Fill a bubble to check yourself
A Correct answer
Multiple choice

A sphere of constant radius 'K' passes through origin and meets axes in A,B,C. The centroid of the $\triangle ABC$ lies on the sphere
  1. $9 (x^{2}+y^{2}+z^{2})=4K^{2}$
  2. $3 (x^{2}+y^{2}+z^{2})=4K^{2}$
  3. $(x^{2}+y^{2}+z^{2})=4K^{2}$
  4. $x^{2}+y^{2}-z^{2}=9K^{2}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let the sphere be x^2 + y^2 + z^2 = K^2. The intercepts are A(K,0,0), B(0,K,0), and C(0,0,K). The centroid is (K/3, K/3, K/3). Substituting this into the sphere equation: (K/3)^2 + (K/3)^2 + (K/3)^2 = 3(K^2/9) = K^2/3. This does not match the options provided; however, the centroid coordinates satisfy 9(x^2+y^2+z^2) = 9(3 * K^2/9) = 3K^2.

Multiple choice

Consider the following statements: 1. The opposite angles of a cyclic quadrilateral are supplementary. 2. Angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle. Which one of the following is correct in respect of the above statements?
  1. $Statement-1\Rightarrow statement-2$
  2. $Statement-2\Rightarrow statement-1$
  3. Neither $Statement-1\Rightarrow statement-2$ nor $Statement-2\Rightarrow statement-1$

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

Statement 2 (angle at center is double the angle at circumference) is a fundamental theorem that can be used to prove Statement 1 (opposite angles of cyclic quadrilateral are supplementary).