Let ABC and Aandrsquo;Bandrsquo;Candrsquo; be two triangles in which AB andgt; Aandrsquo;Bandrsquo;, BC andgt; Bandrsquo;Candrsquo; and CA andgt; Candrsquo;Aandrsquo;. Let D, E and F be the middle points of the sides BC, CA and AB respectively. Let Dandrsquo;, Eandrsquo; and Fandrsquo; be the midpoints of the sides of the sides Bandrsquo;Candrsquo;, Candrsquo;Aandrsquo; and Aandrsquo;Bandrsquo; respectively. Consider the following statements.Statement I:AD andgt; Aandrsquo;Dandrsquo;, BE andgt; Bandrsquo;Eandrsquo; and CF andgt; Candrsquo;Fandrsquo; are always true.Statement II:(AB2+ BC2+ CA2) / (AD2+ BE2+ CF2) = (Aandrsquo;Bandrsquo;2+ Bandrsquo;Candrsquo;2+ Candrsquo;Aandrsquo;2) / (Aandrsquo;Dandrsquo;2+ Bandrsquo;Eandrsquo;2+ Candrsquo;Fandrsquo;2)Q. Which one of the following is correct in respect of the above statements?a)Both Statement I and Statement II are true and Statement II is the correct explanation of Statement I.b)Both Statement I and Statement II are true and Statement II is not the correct explanation of Statement I.c)Statement I is true but Statement II is false.d)Statement I is false but Statement II is true.Correct answer is option 'D'. Can you explain this answer?

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Statement I:
AD > A’D’, BE > B’E’ and CF > C’F’ are always true.

Statement II:
(AB² + BC² + CA²) / (AD² + BE² + CF²) = (A’B’² + B’C’² + C’A’²) / (A’D’² + B’E’² + C’F’²)

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

a)
Both Statement I and Statement II are true and Statement II is the correct explanation of Statement I.
b)
Both Statement I and Statement II are true and Statement II is not the correct explanation of Statement I.
c)
Statement I is true but Statement II is false.
d)
Statement I is false but Statement II is true.

Correct answer is option 'D'. Can you explain this answer?

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CDS II - Mathematics Previous Year Question Paper 2016

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Let ABC and A’B’C’ be two triangles in which AB >...

STATEMENT II:

In ΔADB

AD² + BD² - 2 × AD × BD × cosθ = AB² [∵ Cosine formula]

⇒ AD² + BC²/4 - AD × BC × cosθ = AB² ---- 1 [∵ BD = BC/2]

In ΔADC

AD² + CD² + 2 × AD × CD × cosθ = AC² [∵ Cosine formula, cos(180 - θ = -cosθ]

⇒ AD² + BC²/4 + AD × BC × cosθ = AC² ---- 2 [∵ CD = BC/2]

On adding eq 1 and eq 2.

⇒ 2AD² + BC²/2 = AB² + AC²

⇒ AD² = (2AB² + 2AC² - BC²) /4

Using similar algorithm

⇒ BE² = (2AB² + 2BC² - AC²) /4

⇒ CF² = (2BC² + 2AC² - AB²) /4

⇒ AD² + BE² + CF² = 3(AB² + BC² + CA²) /4

⇒ (AB² + BC² + CA²) / (AD² + BE² + CF²) = 4/3 for any triangle

⇒ Statement II is true

Statement I:

AD² = (2AB² + 2AC² - BC²) /4

Since there is a negative term so no general conclusion can be made. In obtuse angled triangle, statement I will not be true

Let ABC be a triangle such that AB = BC = 5 cm and ∠ABC = 150°

Using cosine formula

⇒ AC = √(AB² + BC² - 2AB × BC cos 150°)

⇒ √(5² + 5² - 2 × 5 × 5 cos 150°) ≈ 9.69

⇒ BE² = (2AB² + 2BC² - AC²) /4

Substituting AB = BC = 5 cm and AC = 9.69 cm

⇒ BE ≈ 1.24 cm

Now A’B’C’ be an equilateral triangle of sides 4 cm

Median of equilateral triangle = √3/2 × side = 2√3 cm

Although all sides of A’B’C’ are less than ABC, its median is greater than the latter’s median

⇒ Statement I is false

∴ Statement I is false but Statement II is true.

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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 middle points of the sides BC, CA and AB respectively. Let D’, E’ and F’ be the midpoints of the sides of the sides B’C’, C’A’ and A’B’ respectively. Consider the following statements.Statement I:AD > A’D’, BE > B’E’ and CF > C’F’ are always true.Statement II:(AB2+ BC2+ CA2) / (AD2+ BE2+ CF2) = (A’B’2+ B’C’2+ C’A’2) / (A’D’2+ B’E’2+ C’F’2)Q. Which one of the following is correct in respect of the above statements?a)Both Statement I and Statement II are true and Statement II is the correct explanation of Statement I.b)Both Statement I and Statement II are true and Statement II is not the correct explanation of Statement I.c)Statement I is true but Statement II is false.d)Statement I is false but Statement II is true.Correct answer is option 'D'. Can you explain this answer?

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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 middle points of the sides BC, CA and AB respectively. Let D’, E’ and F’ be the midpoints of the sides of the sides B’C’, C’A’ and A’B’ respectively. Consider the following statements.Statement I:AD > A’D’, BE > B’E’ and CF > C’F’ are always true.Statement II:(AB2+ BC2+ CA2) / (AD2+ BE2+ CF2) = (A’B’2+ B’C’2+ C’A’2) / (A’D’2+ B’E’2+ C’F’2)Q. Which one of the following is correct in respect of the above statements?a)Both Statement I and Statement II are true and Statement II is the correct explanation of Statement I.b)Both Statement I and Statement II are true and Statement II is not the correct explanation of Statement I.c)Statement I is true but Statement II is false.d)Statement I is false but Statement II is true.Correct answer is option 'D'. Can you explain this answer? for Defence 2024 is part of Defence preparation. The Question and answers have been prepared according to the Defence exam syllabus. Information about 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 middle points of the sides BC, CA and AB respectively. Let D’, E’ and F’ be the midpoints of the sides of the sides B’C’, C’A’ and A’B’ respectively. Consider the following statements.Statement I:AD > A’D’, BE > B’E’ and CF > C’F’ are always true.Statement II:(AB2+ BC2+ CA2) / (AD2+ BE2+ CF2) = (A’B’2+ B’C’2+ C’A’2) / (A’D’2+ B’E’2+ C’F’2)Q. Which one of the following is correct in respect of the above statements?a)Both Statement I and Statement II are true and Statement II is the correct explanation of Statement I.b)Both Statement I and Statement II are true and Statement II is not the correct explanation of Statement I.c)Statement I is true but Statement II is false.d)Statement I is false but Statement II is true.Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for Defence 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for 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 middle points of the sides BC, CA and AB respectively. Let D’, E’ and F’ be the midpoints of the sides of the sides B’C’, C’A’ and A’B’ respectively. Consider the following statements.Statement I:AD > A’D’, BE > B’E’ and CF > C’F’ are always true.Statement II:(AB2+ BC2+ CA2) / (AD2+ BE2+ CF2) = (A’B’2+ B’C’2+ C’A’2) / (A’D’2+ B’E’2+ C’F’2)Q. Which one of the following is correct in respect of the above statements?a)Both Statement I and Statement II are true and Statement II is the correct explanation of Statement I.b)Both Statement I and Statement II are true and Statement II is not the correct explanation of Statement I.c)Statement I is true but Statement II is false.d)Statement I is false but Statement II is true.Correct answer is option 'D'. Can you explain this answer?.

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