Xavier Aptitude Test 2021 Paper

Question : 19 of 100

Marks: +1, -0

$\ABC$ is a triangle with integer-valued sides $\AB = 1$ , $\BC >1$ , and $\CA >1$ . If $\D$ is the mid-point of $\AB$ , then, which of the following options is the closest to the maximum possible value of the angle $\ACD$ (in degrees )?

Validate

Solution:

We will try to maximize the value of the angle ACD:

For a fixed triangle

∠ \ABC

, the

\ACD

can be maximized when we take the median

\CD

to be perpendicular to

\AB

and the value of

\AC

isas small as possible, so that the sine of

∠ \ACD

and hence, the

∠ \ACD

itself if maximized, as the value of

\AD

is fixed at half of

\AB

0.5

Now, the least possible value of

\AC

2

. The triangle will be of sides

(1,2,2)

The value

\Sin (\ACD) = 0.5/2 = 0.25

∠ \ACD = \sin^{-1} (0.25) = 14.78 = 15

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