CAT 2017 Set-1 | Question: 86

Question

Let $\text{ABC}$ be a right-angled triangle with $\text{BC}$ as the hypotenuse. Lengths of $\text{AB}$ and $\text{AC}$ are $15$ km and $20$ km, respectively. The minimum possible time, in minutes, required to reach the hypotenuse from $\text{A}$ at a speed of $30$ km per hour is

• A. $23$
• B. $22$
• C. None of these
• D. $24$ 

Answer 1

Given that, $ABC$ be a right-angled triangle with $BC$ as the hypotenuse.

Also, $AB=15$ km and $AC=20$ km.

We can draw the diagram.



By using the Pythagorean theorem,

$(BC)^{2} = (AB)^{2} + (AC)^{2}$

$\Rightarrow$  $ (BC)^{2} = (15)^{2} + (20)^{2}$

$\Rightarrow$  $ (BC)^{2} = 225 + 400 $

$\Rightarrow$  $ (BC)^{2} = 625  $

$\Rightarrow$  $ (BC)^{2} = (25)^{2}  $

$\Rightarrow$  $ \boxed{BC = 25\;\text{km}} $

The area of the $\triangle  ABC,$ we can write in two ways.

$\frac{1}{2}\times AB\times AC = \frac{1}{2}\times BC\times AD$

 $\Rightarrow 15\times 20 = 25 \times AD $

$\Rightarrow \boxed{AD = 12\;\text{km}}$

$\therefore$  The required time $= \dfrac{12\;\text{km}}{30 \;\frac{\text{km}}{\text{hour}}} = \dfrac{12 }{30 }\times 60$ minute $= 24$ minutes.

Correct Answer $ : \text{D}$