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UPSC CDS 1 2018 Math Paper

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Question : 97 of 100

Marks: +1, -0

In the figure given below, p, q, r are parallel lines; I and m are two transversals.

consider the following :

1. AB : AC = DE : DF

2. AB X EF = BC X DE

Which of the above is/are correct ?

1 only
2 only
Both 1 and 2
Neither 1 nor 2

Solution:

ΔAOB ~ ΔACF (By AA similarity)

\therefore \frac{AB}{AC} = \frac{AO}{AF}

....(1)

Similarly,ΔAFD ~ ΔOFE

\therefore \frac{OF}{AF} = \frac{EF}{DF}

\Rightarrow \frac{AF-AO}{AF} = \frac{DF-DE}{DF}

\Rightarrow \frac{AO}{AF} = \frac{DE}{DF}

....(2)

\frac{AB}{AC} = \frac{DE}{DF}

Hence, statement 1 is correct.

Now,

\frac{AB}{AC} = \frac{DE}{DF}

\Rightarrow \frac{AC}{AB} = \frac{DF}{DE}

\Rightarrow \frac{AB+BC}{AB}

= \frac{DE+EF}{DE}

\Rightarrow \frac{BC}{AB} = \frac{EF}{DE}

\Rightarrow AB \times EF

= BC \times DE

Hence, statement 2 is also correct

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