Question Details

What is the shortest distance between point R and point P?

Study the following information carefully and answer the questions given below:

Rohan starts walking from point W in the east direction after walking 3m he reaches at point P. From there he takes a right turn and walks 2m to reach at point V. From point V he takes a left turn and walks 5m to reach at point S. From point S he takes left turn and walks point 7m to reach at point X. Karan starts walking from point Q in south direction after walking 5m he reaches at point U. From there he takes a right turn and walks 6m to reach at point R. From there he takes again right turn and walks 12m to reach at point T. Now, from point T he takes a left turn and walks 4m to reach at point X.

Options

$\sqrt{130}$ m

130m

$\sqrt{165}$ m

165m

Show Answer

Correct Answer :

Option A

$\sqrt{130}$ m

Solution :

The correct answer is $\sqrt{130}$ m.

Let's map out the positions of the points mentioned in the problem step-by-step using a standard 2D Cartesian coordinate system, where the East direction corresponds to the positive x-axis, West to the negative x-axis, North to the positive y-axis, and South to the negative y-axis.

1. Rohan's Path:
Let us place the starting point W at the origin:
$W = (0, 0)$
Rohan walks 3m East to reach point P:
$P = (3, 0)$
From P, he turns right (South) and walks 2m to reach point V:
$V = (3, - 2)$
From V, he turns left (East) and walks 5m to reach point S:
$S = (3 + 5, - 2) = (8, - 2)$
From S, he turns left (North) and walks 7m to reach point X:
$X = (8, - 2 + 7) = (8, 5)$

2. Karan's Path:
We need to trace Karan's path backwards starting from the known point X. Let the coordinates of Karan's points be represented as variables.
Now, from point T, Karan takes a left turn and walks 4m to reach point X (8, 5). We need to determine the orientation of T relative to X.
Let's trace Karan's path forward:

Karan starts at Q and walks 5m South to U.
From U, he turns right (West) and walks 6m to R.
From R, he turns right (North) and walks 12m to T.
From T (facing North), he turns left (West) and walks 4m to X.

Since turning left from a Northward direction leads West, Karan walks 4m West from T to reach X. This means T is 4m East of X:

T = (8 + 4, 5) = (12, 5)

Working backwards from T to R:
Karan walked 12m North from R to reach T. Therefore, R is 12m South of T:

R = (12, 5 - 12) = (12, - 7)

3. Shortest Distance between R and P:
We have the coordinates of P and R:
$P = (3, 0)$
$R = (12, - 7)$
Using the distance formula, we calculate the shortest distance between P and R:
$Distance = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}}$
$Distance = \sqrt{{(12 - 3)}^{2} + {(- 7 - 0)}^{2}}$
$Distance = \sqrt{9^{2} + {(- 7)}^{2}}$
$Distance = \sqrt{81 + 49}$
$Distance = \sqrt{130} m$