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

Let $a$ and $b$ represent the length of a right triangle's legs. If $d $ is the diameter of a circle inscribed into the triangle and $ D$ is the diameter of a circle circumscribed on the triangle, then $d + D$ equals
  1. $a + b$
  2. $2(a + b)$
  3. $\dfrac{1}{2} (a+b)$
  4. $\sqrt{a^2 + b^2}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

For a right triangle with legs a, b and hypotenuse c = sqrt(a^2 + b^2), the inradius r = (a+b-c)/2 and circumradius R = c/2. Diameter d = 2r = a+b-c. Diameter D = 2R = c. d + D = a+b-c+c = a+b.

Multiple choice

If OA and OB are equal perpendicular chords of the circles $x^2 + y^2 - 2x + 4y = 0$, then equation of OA and OB are where O is origin.
  1. $3x + y = 0$ and $ 3x - y = 0$
  2. $3x + y = 0$ or $3y - x = 0$
  3. $x + 3y = 0$ and $ y - 3x = 0$
  4. $x + y = 0$ or $x - y = 0$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The circle is x^2 + y^2 - 2x + 4y = 0, which is (x-1)^2 + (y+2)^2 = 5. The center is (1, -2). Chords OA and OB are perpendicular and equal, passing through the origin. The lines must be symmetric with respect to the line connecting the origin to the center. Testing option C: x + 3y = 0 and y - 3x = 0 are perpendicular (slopes -1/3 and 3).

Multiple choice

Consider $L_{1}: 2x + 3y +p -3 = 0 L_{2}: 2x + 3y + p + 3 = 0,$ where p is a real number, and $C:x^{2}+y^{2}+6x-10y+ 30= 0$ Statement-1: If line $L_{1}$ is a chord of circle $C$, then $L_{2}$ is not always a diameter of circle $C.$

  1. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

  2. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

  3. Assertion is correct but Reason is incorrect

  4. Both Assertion and Reason are incorrect
Reveal answer Fill a bubble to check yourself
C Correct answer
Multiple choice

A triangle is inscribed in a circle. The vertices of the triangle divide the circle into three arcs of length 3, 4 and 5 units. Then area of the triangle is equal to
  1. $\displaystyle \frac{9\sqrt{3}(1+\sqrt{3})}{\pi ^{2}}$
  2. $\displaystyle \frac{9\sqrt{3}(\sqrt{3}-1)}{\pi ^{2}}$
  3. $\displaystyle \frac{9\sqrt{3}(1+\sqrt{3})}{2\pi^{2}}$
  4. $\displaystyle \frac{9\sqrt{3}(\sqrt{3-1})}{2 \pi^{2}}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The arcs are 3, 4, 5. Total circumference = 12. Radius R = 12 / (2*pi) = 6/pi. The angles subtended by the arcs are 3/12 * 360 = 90 degrees, 4/12 * 360 = 120 degrees, 5/12 * 360 = 150 degrees. Area = 1/2 * R^2 * (sin(90) + sin(120) + sin(150)) = 1/2 * (36/pi^2) * (1 + sqrt(3)/2 + 1/2) = 18/pi^2 * (3/2 + sqrt(3)/2) = 9/pi^2 * (3 + sqrt(3)) = 9*sqrt(3)/pi^2 * (sqrt(3)+1).

Multiple choice

A point $P$ is outside a circle at a distance of $13\;cm$ from its centre. A secant from $P$ cuts the circle in $Q$ and $R$ such that $QR=7\;cm$ and the segment $PQ$ of the secant, exterior to the circle is $9\;cm$. Therefore, the radius of circle is
  1. $3\;cm$
  2. $4\;cm$
  3. $5\;cm$
  4. $6\;cm$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let the secant be PQR where PQ=9 and QR=7. The whole secant PR = 16. Power of point P = PQ * PR = 9 * 16 = 144. Also, Power = d^2 - r^2, where d=13. 144 = 13^2 - r^2 = 169 - r^2. r^2 = 169 - 144 = 25. r = 5.

Multiple choice

Chord is drawn to the circle $\displaystyle x^{2}+y^{2}-4x-2y=0$ at a point where it cuts the x-axis whose slope is parallel to the tangent at an origin. The intercept of the chord on y-axis is
  1. $2$
  2. $4$
  3. $6$
  4. $8$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The circle is (x-2)^2 + (y-1)^2 = 5. It cuts the x-axis at (0,0) and (4,0). The tangent at the origin (0,0) has a slope found by differentiating the circle equation. The chord parallel to this tangent passing through the other point on the x-axis is calculated to find the y-intercept.

Multiple choice

Two circles of radii $10$ cm and $8$ cm intersects each other and the length of the common chord is $12$ cm, find the distance between their centers.
  1. $2$ cm
  2. $(8+ 2\sqrt 7)$ cm
  3. $8$ cm
  4. $2 \sqrt 7$ cm
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let centers be O1, O2. Chord length = 12, so half-chord = 6. Distance from center to chord = sqrt(10^2 - 6^2) = 8 for the 10cm circle, and sqrt(8^2 - 6^2) = sqrt(28) = 2*sqrt(7) for the 8cm circle. Distance between centers = 8 + 2*sqrt(7).

Multiple choice

Determine the length of the chord common to the circles $\displaystyle x^{2}+y^{2}=64 and x^{2}+y^{2}-16x=0$
  1. $2\sqrt{3}$
  2. $4\sqrt{3}$
  3. $6\sqrt{3}$
  4. $8\sqrt{3}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Circle 1: x^2 + y^2 = 64 (center 0,0, r=8). Circle 2: x^2 - 16x + y^2 = 0 -> (x-8)^2 + y^2 = 64 (center 8,0, r=8). The radical axis (common chord) is x^2 + y^2 - 64 - (x^2 - 16x + y^2) = 0, which is 16x - 64 = 0, so x = 4. Substitute x=4 into x^2 + y^2 = 64: 16 + y^2 = 64, y^2 = 48, y = +/- sqrt(48) = +/- 4*sqrt(3). Length = 4*sqrt(3) - (-4*sqrt(3)) = 8*sqrt(3).

Multiple choice

In a circle with centre $P$, chord $AB$ is smaller than the sum of the side $PA$ and side $PB$ by $4$cm. M is the midpoint of AB. If perimeter of $\triangle PAB$ is $144$ cm, then find the length of side $PM$.
  1. $12$ cm
  2. $13$ cm
  3. $11$ cm
  4. $10$ cm
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let PA = PB = r. The perimeter of triangle PAB is PA + PB + AB = 2r + AB = 144. We are given AB = (PA + PB) - 4 = 2r - 4. Substituting this, 2r + (2r - 4) = 144, so 4r = 148, r = 37. In right triangle PAM, PM^2 + AM^2 = PA^2. Since AM = AB/2 = (2r - 4)/2 = r - 2 = 35, PM = sqrt(37^2 - 35^2) = sqrt((37-35)(37+35)) = sqrt(2 * 72) = 12.

Multiple choice

Two parallel chords on the same side of the centre of a circle are $10\ cm$ and $24\ cm$ long and their distance apart is $7\ cm$. Calculate the radius of the circle.
  1. $13\ cm$
  2. $8\ cm$
  3. $5\ cm$
  4. $7\ cm$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let the radius be r and distances from center be x and x+7. Using Pythagoras: r^2 = 5^2 + x^2 and r^2 = 12^2 + (x+7)^2. Equating: 25 + x^2 = 144 + x^2 + 14x + 49. 14x = -168, x = -12. Since distance is absolute, x=5. r^2 = 5^2 + 12^2 = 169, so r=13.