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

$TP$ and $TQ$ are two tangents to a circle $C(0, r)$ at points $P$ and $Q$ respectively. If $\angle POQ = 70^{\circ} $. Then find $\angle PTQ.$
  1. $90$ $^{\circ}$
  2. $70$ $^{\circ}$
  3. $60$ $^{\circ}$
  4. $110$ $^{\circ}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

In the quadrilateral OQTP, the angles at P and Q are 90 degrees because tangents are perpendicular to the radius. The sum of angles in a quadrilateral is 360 degrees. Thus, angle PTQ = 360 - 90 - 90 - 70 = 110 degrees.

Multiple choice

Two concentric circles are such that the smaller divides the larger circle into two regions of equal areas. If the radius of the smaller circle is $2$, then the length of a tangent from any point p on the larger circle to the smaller circle is
  1. $\sqrt{2}$
  2. $2$
  3. $1$
  4. $2$ $\sqrt{2}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let R be radius of larger circle, r=2 be radius of smaller. Area of smaller = pi*r^2 = 4pi. Area of larger = 2 * 4pi = 8pi. So pi*R^2 = 8pi, R^2 = 8, R = 2*sqrt(2). A tangent from a point on the larger circle to the smaller circle forms a right triangle with the radius of the smaller circle and the distance from center to point (R). Length^2 = R^2 - r^2 = 8 - 4 = 4. Length = 2.

Multiple choice

The length of the tangent drawn from any point on the circle ${ x }^{ 2 }+{ y }^{ 2 }+2gx+2fy+p=0$ to the circle ${ x }^{ 2 }+{ y }^{ 2 }+2gx+2fy+q=0$ is:
  1. $\sqrt { q-p }$
  2. $\sqrt { p-q }$
  3. $\sqrt { q+p }$
  4. $\sqrt { 2q+p }$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The length of the tangent from a point (x1, y1) on circle S1 to circle S2 is given by sqrt(S2(x1, y1)). For the given circles, the power of a point on the first circle is the difference between the constant terms. Substituting the point into the second circle equation yields q - p.

Multiple choice

A line meets the coordinate axes in $A$ and $B$. A circle is circumscribed about the triangle $OAB$. If the distance from $A$ and $B$ of the tangent to the circle at the origin be $m$ and $n$, then the diameter of the circle is
  1. $m(m+n)$
  2. $m+n$
  3. $n(m+n)$
  4. ${ m }^{ 2 }+{ n }^{ 2 }$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

For a line x/a + y/b = 1, the distance from origin to the line is d = 1/sqrt(1/a^2 + 1/b^2). The tangent at origin to the circle circumscribing OAB is not standard; however, using geometry, the diameter of the circle is the hypotenuse AB = sqrt(a^2 + b^2). The given m and n relate to the projection of the origin on the axes, leading to the diameter being m + n.

Multiple choice

If the angle between the tangent from $(0,0)$ to the circle ${x}^{2}+{y}^{2}+10x+10y+40=0$ is ${\tan}^{-1}(m)$, then $m=$
  1. $\dfrac{3}{4}$
  2. $\dfrac{1}{2}$
  3. $2$
  4. $\dfrac{4}{3}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The circle is (x+5)^2 + (y+5)^2 = 10. Distance from origin to center is sqrt(50). Tangent length L = sqrt(d^2 - r^2) = sqrt(50 - 10) = sqrt(40). The angle theta between tangents is given by tan(theta/2) = r/L = sqrt(10)/sqrt(40) = 1/2. The question asks for tan(theta) = 2 * tan(theta/2) / (1 - tan^2(theta/2)) = 2 * (1/2) / (1 - 1/4) = 1 / (3/4) = 4/3.