TRIANGLES (CDS)

Total Questions: 69

41. In a APQR, point X is on PQ and point Y is on PR such that XP = 1.5 units, XQ = 6 units, PY = 2 units and YR = 8 units. Which of the following are correct? [2016 (I) Morning Shift]

1. QR = 5XY
2. QR is parallel to XY.
3. APYX is similar to APRQ.

Select the correct answer using the codes given below.

Correct Answer: (d) 1, 2 and 3
Solution:

42. ABC is a triangle in which D is the mid-point of BC and E is the mid-point of AD. Which of the following statement(s) is/are correct? [2016 (I) Morning Shift]

1. The area of ∆ABC is equal to four times the area of ∆BED.
2. The area of ∆ADC is twice the area of ∆BED.

Select the correct answer using the codes given below.

Correct Answer: (c) Both 1 and 2
Solution:In ∆ABC, AD is the median and median AD bisects the area of ∆ABC. Area of ∆ABD = 1/2 ×  Area of ∆ABC

43. ABC is a triangle right angled at C as shown below. Which one of the following is correct? [2016 (I) Morning Shift]

Correct Answer: (c)
Solution:In the given triangle,

44. In the given figure, AD = CD = BC. What is the value of ∠CDB ? [2016 (I) Morning Shift]

Correct Answer: (b) 64°
Solution:Let ∠CDB = y and ∠CAD = x, As, AD = CD

45. ABC is an equilateral triangle and X, Y and Z are the points on BC, CA and AB respectively, such that BX = CY = AZ. Which of the following is/are correct? [2016 (I) Morning Shift]

1. XYZ is an equilateral triangle.
2. ∆XYZ is similar to ∆ABC.

Select the correct answer using the codes given below.

Correct Answer: (c) Both 1 and 2
Solution:In an equilateral ∆ABC, AB = BC = CA and ∠B = ∠C = ∠A

46. A square is inscribed in a right angled triangle with legs p and q and has a common right angle with the triangle. The diagonal of the square is given by [2016 (I) Morning Shift]

Correct Answer: (c)
Solution:In ∆ABC and ∆ADE,


47. In a ∆ABC, if A - B = π/2, then C + 2B is equal to [2016 (I) Morning Shift]

Correct Answer: (d) π/2
Solution:

48. Let ABC be a triangle in which AB = AC . Let L be the locus of points X inside or on the triangle such that BX = CX. Which of the following statements are correct? [2016 (I) Morning Shift]

1. L is a straight line passing through A and incentre of ∆ ABC is on L.
2. L is a straight line passing through A and orthocentre of ∆ ABC is a point on L.
3. L is a straight line passing through A and centroid of ∆ABC is a point on L.

Select the correct answer using the codes given below.

Correct Answer: (d) 1, 2 and 3
Solution:In ∆ABC,


AB AC = i.e. ∆ABC is isosceles triangle. If L is the locus of points X inside or on the triangle, such that BX CX = . Then, L is a straight line passing through A. As, AX is bisecting the line BC, then it passes through the centroid.
As, AB AC = in ∆ABC, then AX is perpendicular to BC, hence it passes through the orthocentre.
In ∆ABC, AB AC = and AX BC ⊥ . So, AX is angle bisector of ∠A, hence it passes through the incentre. Hence, all statements are correct.

49. Suppose ABC is a triangle with AB of unit length. D and E are the points lying on AB and AC respectively, such that BC and DE are parallel. If the area of ∆ ABC is twice the area of ∆ ADE, then the length of AD is [2016 (I) Morning Shift]

Correct Answer: (c) 1/√2 unit
Solution:In ∆ABC, DE is parallel to BC.

50. Let ∆ABC and ∆DEF be such that ∠ABC = ∠DEF, ∠ACB = ∠DFE and ∠BAC = ∠EDF. Let L be the mid-point of BC and M be the mid-point of EF. [2016 (I) Morning Shift]

Consider the following statements
I. ∆ABL and ∆DEM are similar.
II. ∆ALC is congruent to ∆DMF even, if AC ≠ DF .

Which one of the following is correct in respect of the above statements? Codes

Correct Answer: (c) Statement I is true but Statement II is false
Solution:Given that, ∠ABC = ∠DEF, ∠ACB = ∠DFE and ∠BAC = ∠EDF