Let ABC and A prime B prime C prime be two triangles in which AB gt AB prime BC gt B prime C prime a...........

Quantitative Aptitude (CDS) Coordinate Geometry

Let ABC and A′B′C′ be two triangles in which AB > AB′, 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 E′ be the midpoints of he sides B′C′, C′A′ and A′B′ respectively.

Consider the following statement:

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

Statement II. $\small \frac{AB^{2}+BC^{2}+CA^{2}}{AD^{2}+BE^{2}+CF^{2}}=\frac{A'B'^{2}+B'C'^{2}+C'A'^{2}}{A'D'^{2}+B'E'^{2}+C'F'^{2}}$

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

Both Statement I and Statement II are true and Statement II are true and Statement II is the correct explanation of Statement I.

Both Statement I and Statement II are true but Statement II is not the correct explanation of Statement I

Statement I is true but Statement II is false

Statement I is false but Statement II is true

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Explanation:

Since AB > A’B’, BC > B’C’ and CA > C’A’
Also D, E, F are mid points of BC, CA and AB.
Also D’E’F’ are mid points of B’C’, C’A’ and A’B’
Now AD =1/2BC , BE =1/2CA, CF =1/2AB..................... . . (1)
Also A’D’ =1/2B’ C’, B’E’ =1/2C’A’, C’F’ =1/2A’B’............. . . (2)
⇒ If AB > A’B’. BC > B’C’ and CA > C’A’
⇒ A’D’ > AD, BE > B’E’ and CF > C’F’ C’A’
From (1) & (2)
Statement II

⇒ 4 = 4 ⇒ Statement II is true.