text: ## GEOMETRY SET II MCQs
  
  Total Questions : 60 | Page 1 of 6 pages
  Question 1. Suresh is standing on vertex A of triangle ABC, with AB = 3, BC = 5, and CA = 4. Suresh walks according to the following plan: He moves along the altitude-to-the-hypotenuse until he reaches the hypotenuse. He has now cut the original triangle into two triangles; he now walks along the altitude to the hypotenuse of the larger one. He repeats this process forever. What is the total distance that Suresh walks?
  1.    4825
  2.    125
  3.    12
  4.    15
  :
  C
  
  Let M be the end point of the altitude on the hypotenuse. Since, we are dealing with right angle
  triangles, ΔMAC ~ΔABC, so AM = 125. Let N be the endpoint he reaches on side AC. Δ
  MAC ~Δ NAM, So, MNAM =45. This means that each altitude that he walks gets shorter
  by a factor of 45. The total distance is thus an infinite G.P. =[a1r] = 12
  Question 2. There is a ship which is moving away from an iceberg 150 m high, takes 2 mins to change the angle of elevation of the top of the iceberg from 60 to 45. What is the speed of the ship?
  1.    2.46 kmph
  2.    1.3 kmph
  3.    1.9 kmph
  4.    1.7 kmph
  Answer: Option C. -> 1.9 kmph
  :
  C
  
  In a 45-45-90 triangle, sides are in the ratio 1:1:2
  Thus OB= 150
  In a 30-60-90 triangle, sides are in the ration 1:3:2
  Thus OA = 503
  Distance travelled in 2 mins = AB = 150- 503= 63.4 m
  speed = (63.42)× (601000) = 1.9 kmph
  Question 3. All sides of a regular pentagon are extended to form a star with vertices P,Q,R,S,T. What is the sum of the angles made at the vertices?___
  
  :
  
  Interior angle = [(2n-4)x90]/n = 108
  2x= 360-(108x2) = 144
  angle P = 180-144= 36
  Total = 36x5= 180
  Useful to remember: Sum of angles in a star= 180
  Shortcut: Clearly the star point trisects the angle in a pentagon. Hence each angle = 108/3 = 36.
  Sum of angles = 36*5 = 180
  Alternatively: Sum of angles of a n point star = (n – 4) x 180, where n = number of sides of a polygon.
  Here, n = 5.Sum of angles of a n point star = (5 – 4) x 180=180.
  Question 4. Three balls of equal radius are placed such that they are touching each other. A fourth smaller ball is kept such that it touches the other three.Find the ratio of the radii of smaller to larger ball.
  1.    (2−√3)√3
  2.    3- 2√3
  3.    3.5
  4.    4√5
  :
  A
  
  Option (a)
  Let the centers of the large balls be x, y, z and radius R.
  O is the centre of the smaller ball and radius r...
  x, y, z form an equilateral triangle with side equal to 2R.
  O is the centroid of this triangle.
  Therefore ox=oy=oz=R+r= 23(height of the triangle xyz)
  Height=(32)(2R) =3R
  Therefore R+r = 23(3R) rR = (23)3
  Shortcut:- Using the approximation technique used in class, the radius of the bigger circle: smaller circle is close to 0.2.
  Question 5. If O is the centre of a circle and the radius of the circle is 4 units, find the value of length of AC.
  1.    4( √3 + 1)
  2.    9 √2
  3.    10.5
  4.    none of these
  Answer: Option D. -> none of these
  :
  D
  Option (d)
  AC has to less than 8 as the diameter will be of 8 units. So the correct option is “D” as all are greater than 8
  Question 6. What is the maximum number of times x circles of the same size can intersect each other?
  1.    2x
  2.    (x - 1)
  3.    x(x-1)
  4.    x(x+1)/2
  :
  C
  Deducing a Pattern:
  Taking two circles. The number of points of intersection is 2 at most.
  If u consider 3 circles, the number of points of intersection = 6 ( the third circles will have 4 points of intersection with the other 2 circles and those two circles will have 2 points together).
  Shortcut:- Assumption & Reverse Gear:
  Taking two circles. The number of points of intersection is 2 at most. From this itself, option a, b and d can be eliminated giving answer c.
  Question 7. A metal cuboid with sides in the ratio of 2:4:6 is melted to form small cubes of side 2 cm. If the sum of all the edges of the cuboid is 144 cm, then find the ratio of the total surface area of the cuboid to the total surface area of the cubes.
  1.    2:1
  2.    13:4
  3.    11:54
  4.    4:1
  :
  C
  Sides are in the ratio 2:4:6
  Volume of cuboid= 2x .4x . 6x = 48 x3
  Volume of cube= 2.2.2=8
  Number of cubes= 6x3
  Length of all edges of the cuboid= 144 =4(2x+4x+6x) x=3cm
  Ratio = 2[72+108+216]6(27) x 6 x 4
  = 1154
  Question 8. Find the distance between the incentre and the circumcentre of a triangle with circumradius 6 and inradius 2?
  1.    3√2
  2.    2√3
  3.    √10
  4.    None of these
  :
  B
  Use Eulers triangle theorem which states that the distance,d between the incentre and circumcentre of a trinagle is given by d2 = R(R-2r) where R = circum radius, r= inradius
  d2= R(R-2r) = 6(6 - 2 x 2) = 12
  d=23
  Question 9. Two travelers start walking from the same point at an angle of 1500 with each other at the rate of 4 kmph and 3 kmph. Find the distance between them after 2 hours
  1.    2√25+8√3
  2.    28-48√3
  3.    2√25+12√3
  4.    none of these
  :
  C
  option c
  let OA and OB be the paths traveled by the two travelers in 2 hours. Let BCÃ¢Â?Â´ AO at C. then BOC= 180-150= 30
  In right angled triangle OCB, BC= 62=3 KM and OC= 33 km
  In right angled triangle ACB, AB2= AC2 + BC2 = (8+3√3)2+32= 100+483
  AB= 225+123
  Shortcut
  a2+b22abcosα; α is the angle between the paths a and b between the 2 people. 100+48is the answer
  Question 10. With no wastage, small cubes of side 3 inches are formed from a cuboid with the dimensions 2:3:9 are formed. Find the ratio of the body diagonal of the cuboid to that of all of the cubes.
  1.    1:3
  2.    Ratio = √946a23√3
  3.    Ratio = √65a23
  4.    none of these
  Answer: Option B. -> Ratio = √946a23√3
  :
  B
  Number of cubes = 2a×3a×9a3×3×3 = 2a3
  Diagonal of a cube = 4a2+9a2+81a2
  Sum of the diagonals of cubes = 2a3 × 33
  Ratio =4a2+9a2+81a22a3×33
explanation: The document is a set of multiple-choice questions (MCQs) from a geometry test, each followed by an answer and a detailed explanation. The questions cover various topics in geometry, such as triangle properties, circle properties, and spatial relationships. Each question is accompanied by a correct answer and a step-by-step solution.
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distilabel_metadata.content: ## GEOMETRY SET II MCQs
  
  Total Questions : 60 | Page 1 of 6 pages
  Question 1. Suresh is standing on vertex A of triangle ABC, with AB = 3, BC = 5, and CA = 4. Suresh walks according to the following plan: He moves along the altitude-to-the-hypotenuse until he reaches the hypotenuse. He has now cut the original triangle into two triangles; he now walks along the altitude to the hypotenuse of the larger one. He repeats this process forever. What is the total distance that Suresh walks?
  1.    4825
  2.    125
  3.    12
  4.    15
  :
  C
  
  Let M be the end point of the altitude on the hypotenuse. Since, we are dealing with right angle
  triangles, ΔMAC ~ΔABC, so AM = 125. Let N be the endpoint he reaches on side AC. Δ
  MAC ~Δ NAM, So, MNAM =45. This means that each altitude that he walks gets shorter
  by a factor of 45. The total distance is thus an infinite G.P. =[a1r] = 12
  Question 2. There is a ship which is moving away from an iceberg 150 m high, takes 2 mins to change the angle of elevation of the top of the iceberg from 60 to 45. What is the speed of the ship?
  1.    2.46 kmph
  2.    1.3 kmph
  3.    1.9 kmph
  4.    1.7 kmph
  Answer: Option C. -> 1.9 kmph
  :
  C
  
  In a 45-45-90 triangle, sides are in the ratio 1:1:2
  Thus OB= 150
  In a 30-60-90 triangle, sides are in the ration 1:3:2
  Thus OA = 503
  Distance travelled in 2 mins = AB = 150- 503= 63.4 m
  speed = (63.42)× (601000) = 1.9 kmph
  Question 3. All sides of a regular pentagon are extended to form a star with vertices P,Q,R,S,T. What is the sum of the angles made at the vertices?___
  
  :
  
  Interior angle = [(2n-4)x90]/n = 108
  2x= 360-(108x2) = 144
  angle P = 180-144= 36
  Total = 36x5= 180
  Useful to remember: Sum of angles in a star= 180
  Shortcut: Clearly the star point trisects the angle in a pentagon. Hence each angle = 108/3 = 36.
  Sum of angles = 36*5 = 180
  Alternatively: Sum of angles of a n point star = (n – 4) x 180, where n = number of sides of a polygon.
  Here, n = 5.Sum of angles of a n point star = (5 – 4) x 180=180.
  Question 4. Three balls of equal radius are placed such that they are touching each other. A fourth smaller ball is kept such that it touches the other three.Find the ratio of the radii of smaller to larger ball.
  1.    (2−√3)√3
  2.    3- 2√3
  3.    3.5
  4.    4√5
  :
  A
  
  Option (a)
  Let the centers of the large balls be x, y, z and radius R.
  O is the centre of the smaller ball and radius r...
  x, y, z form an equilateral triangle with side equal to 2R.
  O is the centroid of this triangle.
  Therefore ox=oy=oz=R+r= 23(height of the triangle xyz)
  Height=(32)(2R) =3R
  Therefore R+r = 23(3R) rR = (23)3
  Shortcut:- Using the approximation technique used in class, the radius of the bigger circle: smaller circle is close to 0.2.
  Question 5. If O is the centre of a circle and the radius of the circle is 4 units, find the value of length of AC.
  1.    4( √3 + 1)
  2.    9 √2
  3.    10.5
  4.    none of these
  Answer: Option D. -> none of these
  :
  D
  Option (d)
  AC has to less than 8 as the diameter will be of 8 units. So the correct option is “D” as all are greater than 8
  Question 6. What is the maximum number of times x circles of the same size can intersect each other?
  1.    2x
  2.    (x - 1)
  3.    x(x-1)
  4.    x(x+1)/2
  :
  C
  Deducing a Pattern:
  Taking two circles. The number of points of intersection is 2 at most.
  If u consider 3 circles, the number of points of intersection = 6 ( the third circles will have 4 points of intersection with the other 2 circles and those two circles will have 2 points together).
  Shortcut:- Assumption & Reverse Gear:
  Taking two circles. The number of points of intersection is 2 at most. From this itself, option a, b and d can be eliminated giving answer c.
  Question 7. A metal cuboid with sides in the ratio of 2:4:6 is melted to form small cubes of side 2 cm. If the sum of all the edges of the cuboid is 144 cm, then find the ratio of the total surface area of the cuboid to the total surface area of the cubes.
  1.    2:1
  2.    13:4
  3.    11:54
  4.    4:1
  :
  C
  Sides are in the ratio 2:4:6
  Volume of cuboid= 2x .4x . 6x = 48 x3
  Volume of cube= 2.2.2=8
  Number of cubes= 6x3
  Length of all edges of the cuboid= 144 =4(2x+4x+6x) x=3cm
  Ratio = 2[72+108+216]6(27) x 6 x 4
  = 1154
  Question 8. Find the distance between the incentre and the circumcentre of a triangle with circumradius 6 and inradius 2?
  1.    3√2
  2.    2√3
  3.    √10
  4.    None of these
  :
  B
  Use Eulers triangle theorem which states that the distance,d between the incentre and circumcentre of a trinagle is given by d2 = R(R-2r) where R = circum radius, r= inradius
  d2= R(R-2r) = 6(6 - 2 x 2) = 12
  d=23
  Question 9. Two travelers start walking from the same point at an angle of 1500 with each other at the rate of 4 kmph and 3 kmph. Find the distance between them after 2 hours
  1.    2√25+8√3
  2.    28-48√3
  3.    2√25+12√3
  4.    none of these
  :
  C
  option c
  let OA and OB be the paths traveled by the two travelers in 2 hours. Let BCÃ¢Â?Â´ AO at C. then BOC= 180-150= 30
  In right angled triangle OCB, BC= 62=3 KM and OC= 33 km
  In right angled triangle ACB, AB2= AC2 + BC2 = (8+3√3)2+32= 100+483
  AB= 225+123
  Shortcut
  a2+b22abcosα; α is the angle between the paths a and b between the 2 people. 100+48is the answer
  Question 10. With no wastage, small cubes of side 3 inches are formed from a cuboid with the dimensions 2:3:9 are formed. Find the ratio of the body diagonal of the cuboid to that of all of the cubes.
  1.    1:3
  2.    Ratio = √946a23√3
  3.    Ratio = √65a23
  4.    none of these
  Answer: Option B. -> Ratio = √946a23√3
  :
  B
  Number of cubes = 2a×3a×9a3×3×3 = 2a3
  Diagonal of a cube = 4a2+9a2+81a2
  Sum of the diagonals of cubes = 2a3 × 33
  Ratio =4a2+9a2+81a22a3×33
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document is a set of multiple-choice questions (MCQs) from a geometry test, each followed by an answer and a detailed explanation. The questions cover various topics in geometry, such as triangle properties, circle properties, and spatial relationships. Each question is accompanied by a correct answer and a step-by-step solution.
  
  Label: multiple-question-and-answer