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 the lines $3x-4y+4=0$ and $6x-8y-7=0$ are tangents to a circle, then find the radius of the circle.
  1. $3/4$
  2. $4/3$
  3. $1/4$
  4. $7/4$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The lines are 3x - 4y + 4 = 0 and 6x - 8y - 7 = 0. Rewrite the second as 3x - 4y - 3.5 = 0. The distance between parallel lines ax + by + c1 = 0 and ax + by + c2 = 0 is |c1 - c2| / sqrt(a^2 + b^2). Distance = |4 - (-3.5)| / sqrt(3^2 + 4^2) = 7.5 / 5 = 1.5. The diameter is 1.5, so the radius is 0.75 or 3/4.

Multiple choice

Length of the chord joining the points $P(\alpha)$ and $Q(\beta)$ on the circle $x^{2}+y^{2}=a^{2}$ is
  1. $2a \cos \left(\dfrac{\alpha-\beta}{2}\right)$
  2. $2a \sin \left(\dfrac{\alpha-\beta}{2}\right)$
  3. $2a \tan \left(\dfrac{\alpha-\beta}{2}\right)$
  4. $2a \csc \left(\dfrac{\alpha-\beta}{2}\right)$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The distance between two points on a circle with polar angles alpha and beta is given by the chord length formula 2a * sin((alpha - beta) / 2).

Multiple choice

Chord $AB$ of the circle ${x^2} + {y^2} = 100$ passes through the point $\left( {7,1} \right)$ and subtends an angles of ${60^0}$ at the circumference of the circle. If ${m_1}$ and ${m_2}$ are the slopes of two such chords then the value of ${m_1}{m_2}$ is
  1. $-1$
  2. $1$
  3. $\frac{7}{{12}}$
  4. $ - 3$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The angle subtended by a chord at the circumference is half the angle at the center. If it subtends 60 degrees at the circumference, it subtends 120 degrees at the center. The distance from the center (0,0) to the chord is d = R * cos(120/2) = 10 * cos(60) = 5. The line equation passing through (7,1) with slope m is y - 1 = m(x - 7) or mx - y + (1 - 7m) = 0. Distance from origin = |1 - 7m| / sqrt(m^2 + 1) = 5. Squaring: (1 - 7m)^2 = 25(m^2 + 1). 1 - 14m + 49m^2 = 25m^2 + 25. 24m^2 - 14m - 24 = 0. Product of roots m1*m2 = c/a = -24/24 = -1.

Multiple choice

STATEMENT - 1 : The locus of the middle points of equal chords of a circle with centre at O is a circle with centre at O STATEMENT - 2 : The mid point of the equal chords are equidistant from the centre of the circle.

  1. Statement - 1 is True, Statement - 2 is True, Statement - 2 is a correct explanation for Statement - 1

  2. Statement - 1 is True, Statement - 2 is True : Statement 2 is NOT a correct explanation for Statement - 1

  3. Statement - 1 is True, Statement - 2 is False

  4. Statement - 1 is False, Statement - 2 is True
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Equal chords in a circle are equidistant from the center. The locus of points equidistant from the center is a circle. Both statements are true and the second explains the first.

Multiple choice

A square with side length $l$ is inscribed in a semi-circle such that one side of the square is on the diameter of the semicircle. The perimeter of the semicircle is given by:
  1. $l \pi \sqrt{5}$
  2. $\displaystyle \frac{\pi \sqrt{5} l}{2}$
  3. $\sqrt{5} l \left ( \dfrac{\pi}{2} + 1 \right )$
  4. $\displaystyle \frac{\sqrt{5} l }{2} (\pi + 1)$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let the square have side l. The vertices on the semi-circle are at (l/2, l) and (-l/2, l). The radius R satisfies R^2 = (l/2)^2 + l^2 = 5l^2/4, so R = l*sqrt(5)/2. Perimeter of semi-circle = pi*R + 2R = R(pi+2) = (l*sqrt(5)/2)(pi+2) = sqrt(5)*l(pi/2 + 1).

Multiple choice

Two chords AB & CD of lengths $6$ cm & $8$ cm of a circle are parallel and are on the same side of its circle. If the distance between them is $1$ cm. Find the radius of the circle.
  1. $3$ cm
  2. $4$ cm
  3. $5$ cm
  4. $2$ cm
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let the radius be r. Distance from center to chord of length 6 is sqrt(r^2 - 3^2) = sqrt(r^2 - 9). Distance to chord of length 8 is sqrt(r^2 - 4^2) = sqrt(r^2 - 16). The difference is 1. sqrt(r^2 - 9) - sqrt(r^2 - 16) = 1. Testing r=5: sqrt(25-9) - sqrt(25-16) = 4 - 3 = 1. Correct.

Multiple choice

$AB$ is a diameter of the circle, $CD$ is the chord equal to the radius the of the circle. $AC$ and $BD$ when extended intersect at a point $E$. Then find measure of $\angle AEB$.
  1. ${30^ \circ }$
  2. ${60^ \circ }$
  3. ${90^ \circ }$
  4. ${120^ \circ }$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

AB is diameter. CD = radius. Triangle OCD is equilateral (OC=OD=CD=r). Angle COD = 60 degrees. Angle CBD = 1/2 angle COD = 30 degrees (angle at circumference). Angle ACB = 90 degrees (angle in semicircle). In triangle BCE, angle CBE = 30, angle BCE = 180 - 90 = 90. Angle CEB = 180 - 90 - 30 = 60 degrees.

Multiple choice

Let $S=x^{2}+y^{2}+2gx+2fy+c=0$ be a given circle . Then the locus of the foot of the perpendicular drawn from the origin upon any chords of $S$ which subtends right angle at the origin is:
  1. $x^{2}+y^{2}+gx+fy+c/2=0$
  2. $x^{2}+y^{2}=g$
  3. $x^{2}+y^{2}=f$
  4. $x^{2}+y^{2}+g=0$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

If a chord of x^2 + y^2 + 2gx + 2fy + c = 0 subtends a right angle at the origin, the locus of the foot of the perpendicular from the origin to the chord is x^2 + y^2 + gx + fy + c/2 = 0. This is a standard result derived from the equation of the chord of contact and the condition of perpendicularity.