TRIANGLES (CDS)

Total Questions: 69

51. ABC and DEF are similar triangles. If the ratio of side AB to side DE is (√2+1) : √3, then the ratio of area of ∆ ABC to that of ∆DEF is [2016 (I) Morning Shift]

Correct Answer: (c) 1 : (9 - 6√2)
Solution:Given, ∆ABC ~ ∆DEF

52. Let ABC and A' B'C' be two triangles in which AB > A'B', BC > B'C' and CA> C'A'. Let D, E and F be the mid-points of the sides BC, CA and AB, respectively. Let D', E' and F' be the mid-points of the sides B'C', C'A' and A'B', respectively. [2016 (I) Morning Shift]

Correct Answer: (a) Both Statement I and Statement II are true and Statement II is the correct explanation of Statement I
Solution:Given, in ∆ABC and ∆A'B'C',



53. Two poles are placed at P and Q on either side of a road such that the line joining P and Q is perpendicular to the length of the road. A person moves x m away from P parallel to the road and places another pole at R. Then, the person moves further x m in the same direction and turns and moves a distance y m away from the road perpendicularly, where he finds himself, Q and R on the same line. The distance between P and Q (i.e. the width of the road) in metre is [2016 (I) Morning Shift]

Correct Answer: (c) y
Solution:We can draw the figure on the basis of given statements, which is as follows

54. The point O is equidistant from the three sides of a ∆ABC. Consider the following statements: [2015 (II) Evening Shift]

I. ∠OAC + ∠OCB + ∠OBA = 90°
II. ∠BOC = 2 ∠BAC
III. The perpendiculars drawn from any point on OA to AB and AC are always equal

Which of the above statements are correct?

Correct Answer: (c) I and III
Solution:Given, O is equidistant from AB, BC and CA.


55. If the angles of a triangle are in the ratio 4 : 1 : 1. Then, the ratio of the largest side to the perimeter is [2015 (I) Morning Shift]

Correct Answer: (c)
Solution:Let the angles of a triangle be 4x, x and x, respectively.


56. Consider the following statements [2015 (I) Morning Shift]

I. Let D be a point on the side BC of a ∆ABC. If area of ∆ABD = area of ∆ACD, then for all points O on AD, area of ∆ABO = area of ∆ACO.
II. If G is the point of concurrence of the medians of a ∆ABC, then area of ∆ABG = area of ∆BCG = area of ∆ACG.

Which of the above statement(s) is/are correct?

Correct Answer: (c) Both I and II
Solution:



57. In a ∆ABC, AD is the median through A and E is the mid-point of AD and BE produced meets AC at F. Then, AF is equal to [2014 (II) Evening Shift]

Correct Answer: (c) AC/3
Solution:Given, in ∆ ABC, AD is the median and E is the mid-point of AD. Through point D, draw a line l || BF which intersect AC at G.

58. Three straight lines are drawn through the three vertices of a ∆ ABC, the line through each vertex being parallel to the opposite side. Then ∆ DEF is bounded by these parallel lines. [2014 (II) Evening Shift]

Consider the following statements in respect of the ∆ DEF.
I. Each side of ∆DEF is double the side of ∆ ABC to which it is parallel.
II. Area of ∆DEF is four times the area of ∆ ABC.

Which of the above statement(s) is/are correct?

Correct Answer: (c) Both I and II
Solution:


59. In a ∆ ABC, if ∠B = ∠2C = ∠2A . Then, what is the ratio of AC to AB ? [2014 (II) Evening Shift]

Correct Answer: (a) 2 : 1
Solution:Given, in ∆ABC, ∠B = 2∠C = 2∠A We know that, sum of angles of a triangle = 180°

60. Consider the following statements in respect of an equilateral triangle [2014 (II) Evening Shift]

I. The altitudes are congruent.
II. The three medians are congruent.
III. The centroid bisects the altitude.

Which of the above statements are correct?

Correct Answer: (a) I and II
Solution:Since, the altitude and medians of an equilateral triangle are congruent but centroid divide the altitude in 2 :1. So, statements I and II are correct.