Geometry

There are 24 equally spaced points lying on the circumference of a circle. What is the maximum number of equilateral triangles that can be drawn by taking sets of three points as the vertices?

1. 4
2. 6
3. 8
4. 12

How many diagonals can be drawn by joining the vertices of an octagon?

1. 20
2. 24
3. 28
4. 64

Let ABCD be a rectangle. Let P, Q, R, S be the mid-points of sides AB, BC, CD, DA respectively. Then the quadrilateral PQRS is a

1. Square
2. Rectangle, but need not be a square
3. Rhombus, but need not be a square
4. Parallelogram, but need not be a rhombus

Let P, Q, R be the mid-points of sides AB, BC, CA respectively of a triangle ABC. If the area of the triangle ABC is 5 square units, then the area of the triangle PQR is

1. 5/3 square units
2. 5/2√2 square units
3. 5/4 square units
4. 1 square unit

Two parallel chords of a circle whose diameter is 13 cm are respectively 5 cm and 12 cm in length. If both the chords are on the same side of the diameter, then the distance between these chords is

1. 5.5 cm
2. 5 cm
3. 3.5 cm
4. 3 cm

ABC is a triangle and D is a point on the side BC. If BC = 12 cm, BD = 9 cm and ∠ADC = ∠BAC, then the length of AC is equal to

1. 5 cm
2. 6 cm
3. 8 cm
4. 9 cm

In the figure given below, PQRS is a parallelogram. PA bisects angle P and SA bisects angle S. What is angle PAS equal to?

1. 60°
2. 75°
3. 90°
4. 100°

In the figure given below, ∠A = 80° and ∠ABC = 60°. BD and CD bisect angles B and C respectively. What are the values of x and y respectively?

1. 10 and 130
2. 10 and 125
3. 20 and 130
4. 20 and 125

In the figure given below, PQR is a non-isosceles right-angled triangle, right angled at Q. If LM and QT are parallel and QT= PT, then what is ∠RLM equal to?

1. ∠PQT
2. ∠LRM
3. ∠RML
4. ∠QPT

In the figure given below, PQ is parallel to RS and PR is parallel to QS, If ∠LPR = 35° and ∠UST = 70°, then what is ∠MPQ equal to?

1. 55°
2. 70°
3. 75°
4. 80°

In the figure given below, ABC is a triangle with AB = BC and D is an interior point of the triangle ABC such that ∠DAC = ∠DCA.

Consider the following statements:

1. Triangle ADC is an isosceles triangle.
2. D is the centroid of the triangle ABC.
3. Triangle ABD is congruent to the triangle CBD.

Which of the above statements are correct?

1. 1 and 2 only
2. 2 and 3 only
3. 1 and 3 only
4. 1, 2 and 3

In the figure given below, M is the mid-point of AB and ∠DAB = ∠CBA and ∠AMC = ∠BMD. Then the triangle ADM is congruent to the triangle BCM by

1. SAS rule
2. SSS rule
3. ASA rule
4. AAA rule

ABCD is a square. X is the mid-point of AB and Y is the mid-point of BC. Consider the following statements:

1. Triangles ADX and BAY are congruent
2. ∠DXA = ∠AYB
3. DX is inclined at an angle 60° with AY
4. DX is not perpendicular to AY

Which of the above statements are correct?

1. 2, 3 and 4 only
2. 1, 2 and 4 only
3. 1, 3 and 4 only
4. 1 and 2 only

How many degrees are there in an angle which equals two-third of its complement?

1. 60°
2. 48°
3. 45°
4. 36°

In the figure given, ∠B = 38°, AC = BC and AD = CD. What is ∠D equal to?

1. 26°
2. 28°
3. 38°
4. 52°

ABC is an isosceles triangle such that AB = BC = 8 cm and ∠ABC = 90°. What is the length of the perpendicular drawn from B on AC?

1. 4√2 cm
2. 4 cm
3. 2 cm
4. 2√2 cm

What is the length of the perpendicular drawn from the centre of circle of radius r on the chord of length √3r?

1. r/4
2. r/2
3. √2r
4. r

If A, B, C, D are the successive angles of a cyclic quadrilateral, then what is cos A + cos B + cos C + cos D equal to?

1. 0
2. 1
3. 2
4. 4

A circle of radius 10 cm has an equilateral triangle inscribed in it. The length of the perpendicular drawn from the centre to any side of the triangle is

1. 5√3 cm
2. 5 cm
3. 10√3 cm
4. 10 cm

In the figure given, LM is parallel to QR. If LM divides the triangle PQR such that area of trapezium LMRQ is two times the area of triangle PLM, then what is PL/PQ equal to?

1. 1/3
2. 1/√2
3. 1/√3
4. 1/2

The angles of a triangle are in the ratio 2:4:3. The smallest angle of the triangle is

1. 20°
2. 40°
3. 50°
4. 60°

In a circle with center O and radius 1 cm, an arc AB makes an angle 60 degrees at O. Let R be the region bounded by the radii OA, OB and the arc AB. If C and D are two points
on OA and OB, respectively, such that OC = OD and the area of triangle OCD is half that of R, then the length of OC, in cm, is

1. (π/3√3)^1/2
2. (π/4)^1/2
3. (π/6)^1/2
4. (π/4√3)^1/2

In a parallelogram ABCD of area 72 sq cm, the sides CD and AD have lengths 9 cm and 16 cm, respectively. Let P be a point on CD such that AP is perpendicular to CD. Then the area, in sq cm, of triangle APD is

1. 18√3
2. 24√3
3. 32√3
4. 12√3

Let ABCD be a rectangle inscribed in a circle of radius 13 cm. Which one of the following pairs can represent, in cm, the possible length and breadth of ABCD?

1. 24, 10
2. 25, 9
3. 24, 12
4. 25, 10

Points E, F, G, H lie on the sides AB, BC, CD, and DA, respectively, of a square ABCD. If EFGH is also a square whose area is 62.5% of that of ABCD and CG is longer than EB, then the ratio of length of EB to that of CG is

1. 2 : 5
2. 4 : 9
3. 3 : 8
4. 1 : 3

In a circle, two parallel chords on the same side of a diameter have lengths 4 cm and 6 cm. If the distance between these chords is 1 cm, then the radius of the circle, in cm, is

1. √12
2. √14
3. √13
4. √11

A circle is inscribed in a given square and another circle is circumscribed about the square. What is the ratio of the area of the inscribed circle to that of the circumscribed circle?

1. 2 : 3
2. 3 : 4
3. 1 : 2
4. 1 : 4

A semicircle is drawn with AB as its diameter. From C, a point on AB, a line perpendicular to AB is drawn meeting the circumference of the semi-circle at D. Given that AC = 2 cm and CD = 6 cm, the area of the semicircle (in sq. cm) will be:

1. 32 π
2. 40.5 π
3. 50 π
4. 81 π

Consider obtuse angled triangles with sides 8 cm, 15 cm and x cm. If x is an integer then how many such triangles exist?

1. 5
2. 10
3. 15
4. 21

If in the figure below, angle XYZ = 90° and the length of the arc XZ = 10π, then the area of the sector XYZ is

1. 10π
2. 25π
3. 100π
4. None of the above

AB is a chord of a circle. The length of AB is 24 cm. P is the midpoint of AB. Perpendiculars from P on either side of the chord meets the circle at M and N respectively. If PM < PN and PM = 8 cm. then what will be the length of PN?

1. 17 cm
2. 18 cm
3. 19 cm
4. 20 cm
5. 21 cm

AB, CD and EF are three parallel lines, in that order. Let d₁ and d₂ be the distances from CD to AB and EF respectively. d₁ and d₂ are integers, where d₁ : d₂ = 2 : 1. P is a point on AB, Q and S are points on CD and R is a point on EF. If the area of the quadrilateral PQRS is 30 square units, what is the value of QR when value of SR is the least?

1. slightly less than 10 units
2. 10 units
3. slightly greater than 10 units
4. slightly less than 20 units
5. slightly greater than 20 units

The points (2, 5) and (6, 3) are two end points of a diagonal of a rectangle. If the other diagonal has the equation y = 3x + c, then c is

1. -5
2. -6
3. -7
4. -8