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

A rectangular display board has corners A(0, 0), B(60, 0), C(60, 20), and D(0, 20), measured in cm. A circular arc passes through points A and C and meets side AB again at P, where AP = 50 cm. What is the radius of the circle?

  1. 25 cm

  2. 25 2 cm

  3. 50 cm

  4. 45 cm
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Points are A(0,0), B(60,0), C(60,20). A circle passing through (0,0) and (60,20) has its center on the perpendicular bisector. Given AP=50 on the x-axis, P is (50,0). The circle passes through (0,0), (60,20), and (50,0). Solving for the circle equation x^2 + y^2 + Dx + Ey + F = 0 yields the radius.

Multiple choice

On a coordinate map, the circle x2 + y2 = 10mx + 14ny intersects the x-axis at A(α, 0) and the y-axis at B(0, β), where αβ is not equal to 0. If point C(p, q) lies on chord AB, what is the value of ( p + α ) ( 5 m ) + ( q + β ) ( 7 n ) ?

  1. 4

  2. 5

  3. 6

  4. 8
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The nonzero intercepts are alpha = 10m and beta = 14n. Since C lies on AB, p/(10m) + q/(14n) = 1, so (p + alpha)/(5m) + (q + beta)/(7n) = 2 + 2 = 6, making option C correct.

Multiple choice

PQ and QR are the two chords of a circle. Line segment RP is stretched to a point S and then SQ is joined which intersects the circle at T. PQ = QR = 9 cm, QT = 4 cm. What is the measure of SQ?

  1. 18 cm

  2. 20 cm

  3. 21.25 cm

  4. 20.25 cm

  5. a
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

By the Power of a Point theorem for the circle, if RP is extended to S, then SP * SR = ST * SQ. Let SQ = x. Then ST = x - 4. Since PQ = QR = 9, the triangle is isosceles. Using geometry properties, the segment lengths satisfy the power of a point relation, resulting in x = 20.25.

Multiple choice

The perpendicular from the centre of a circle to a chord is 16 cm in length. If the diameter of the circle is 40 cm, then what is the length of the chord?

  1. 12 cm

  2. 16 cm

  3. 24 cm

  4. 30 cm
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The radius of the circle is half the diameter, which is 20 cm. Using the Pythagorean theorem on the triangle formed by the radius, the perpendicular distance (16 cm), and half the chord length, we get sqrt(20^2 - 16^2) = sqrt(400 - 256) = sqrt(144) = 12 cm. The total chord length is twice this value, which is 24 cm.

Multiple choice

A chord PQ of the circle C divides it into two segments such that 3 times the area of the major segment is 4 times the area of the minor segment. Question: What is the radius of C? Statement I: Area of the minor segment is 66 square cm. Statement II: Area of the major segment is 88 square cm.

  1. The Question can be answered by one of the Statements alone but not by the other.

  2. The Question can be answered by either Statement alone.

  3. The Question can be answered by using both the Statements together, but cannot be answered by using either Statement alone.

  4. The Question cannot be answered even by using both Statements together.
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The area of the circle is fixed. If we know the area of either the major or minor segment, we can determine the total area of the circle and thus the radius. Both statements provide sufficient information.

Multiple choice

Directions: The following problem contains a question and two statements giving certain data. You have to decide whether the data given in the statements is sufficient for answering the question. What is the area of the parallelogram inscribed in a circle? (A) The sum of the two sides of the parallelogram is 10 cm. (B) Area of the circle is 75 cm2.

  1. If statement (A) alone is sufficient but statement (B) alone is not.

  2. If statement (B) alone is sufficient but statement (A) alone is not.

  3. If both statements (A) and (B) together are sufficient but none of them alone is sufficient.

  4. If both statements (A) and (B) independently are sufficient.

  5. If both statements (A) and (B) together are not sufficient.
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Area of a parallelogram inscribed in a circle is only possible if it is a rectangle. The diagonal of the rectangle is the diameter of the circle. Statement (B) gives the area of the circle (pi*r^2), which gives the radius and diameter. Statement (A) gives the sum of sides (L+W). With diagonal (D) and L+W, we can find L*W (Area) using (L+W)^2 = L^2+W^2+2LW = D^2+2Area. Both are needed.

Multiple choice

Directions: The following question contains statement - 1 (Assertion), statement - 2 (Reason), 4 choices (A), (B), (C) and (D) out of which only one is correct. Tangents are drawn from the point (17, 7) to the circle x2 + y2 = 169 Statement - 1 The tangents are mutually perpendicular. Because Statement - 2 The locus of the points from which mutually perpendicular tangents can be drawn to the given circle is x2 + y2 = 338.

  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

The locus of points from which mutually perpendicular tangents can be drawn to a circle x^2 + y^2 = r^2 is the director circle x^2 + y^2 = 2r^2. Here r^2 = 169, so 2r^2 = 338. Since (17, 7) satisfies x^2 + y^2 = 17^2 + 7^2 = 289 + 49 = 338, the tangents are indeed perpendicular.

Multiple choice

Passage

To the circle $ x^2+y^2=4$ two tangents are drawn from $P(4, 0)$, which touches the circle at $T_1$ and $T_2$ and a rhombus $PT_1P'T_2$ is completed.

Circumcentre of the triangle $PT_1T_2$ is at
  1. $(2, 0)$
  2. $(0, 2)$
  3. $\left (\displaystyle \frac{\sqrt{3}}{2},0 \right )$

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

The tangents from P(4,0) to x^2 + y^2 = 4 touch at points T1 and T2. The triangle PT1T2 is isosceles. The circumcentre of a triangle formed by tangents and the chord of contact is the midpoint of the line segment connecting the external point and the origin, or specifically calculated via geometry as (2,0).

Multiple choice

Passage

Tangents are drawn from the point $P(1,-1)$ to the circle $x^{2}+y^{2}-4x-6y-3=0$ with centre C. A and B are the points of contact.

The equation of the circle circumscribing the triangle formed by pair of tangent and corresponding chord of contact, is
  1. $x^{2}+y^{2}+3x-2y-1=0$
  2. $x^{2}+y^{2}+3x+2y-1=0$
  3. $x^{2}+y^{2}-3x-2y-1=0$

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

The circle circumscribing the triangle formed by the pair of tangents and the chord of contact is the circle with the line segment CP as diameter, where C is the center of the circle and P is the external point. C = (2, 3), P = (1, -1). Diameter form: (x - 2)(x - 1) + (y - 3)(y + 1) = 0. x^2 - 3x + 2 + y^2 - 2y - 3 = 0. x^2 + y^2 - 3x - 2y - 1 = 0.

Multiple choice

Passage

For points $P\equiv \left({x}{1},{y}{1}\right)$ and $Q\equiv\left({x}{2},{y}{2}\right)$ of the coordinate plane, a new distance $d\left(P,Q\right)=\left|{x}{1}-{x}{2}\right|+\left|{y}{1}-{y}{2}\right|$ Let $O\equiv\left(0,0\right), A\equiv\left(2,3\right)$ and $C\equiv\left(4,3\right)$ are four fixed points on the $x-y$ plane. On the basis of the above information, answer the following questions:

Distance between the circumcentre and orthocentre of $\triangle ABC$ is
  1. $5$
  2. $\sqrt{29}$
  3. $7$
  4. $\sqrt{37}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Multiple choice

Passage

Directions: This data sufficiency problem consists of a question and two statements labelled (1) and (2). You have to decide whether the data given in the statements is sufficient for answering the question.

A triangle ABC has a right angle at C. What is the value of (AC + BC)? (1) The radius of the inscribed circle of triangle ABC is equal to 5 cm. (2) The radius of the circumscribed circle of triangle ABC is equal to 9 cm.

  1. Statement (1) alone is sufficient, but statement (2) alone is not sufficient.

  2. Statement (2) alone is sufficient, but statement (1) alone is not sufficient.

  3. Both statements (1) and (2) together are sufficient, but neither of them alone is sufficient.

  4. Each statement alone is sufficient.

  5. Both statements (1) and (2) together are not sufficient.
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

For a right triangle, the inradius r = (a+b-c)/2 and circumradius R = c/2. Statement 1 gives r=5, so a+b-c = 10. Statement 2 gives R=9, so c=18. Together, a+b-18 = 10, so a+b = 28.

Multiple choice

Passage

Mark option (a) if the question can be answered by using one of the statements alone, but cannot be answered using the other statement alone. Mark option (b) if the question can be answered by using either statement alone. Mark option (c) if the question can be answered by using both the statements together, but cannot be answered using either statement alone. Mark option (d) if the question cannot be answered even by using both the statements together.

Question: What is the area of the triangle inscribed in a semi-circle with the diameter as the base? Statement-I: The diameter of semi-circle is 20 cm. Statement-II: Two shorter sides of the triangle are 12 cm and 16 cm.

  1. (a)

  2. (b)

  3. (c)

  4. (d)
Reveal answer Fill a bubble to check yourself
A Correct answer