Question

Three girls Reshma, Salma and Mandip are playing a game by standing on a circle of radius 5m drawn in a park. Reshma throws a ball to Salma, Salma and Mandip, Mandip to Reshma. If the distance between Reshma and Salma and between Salma and Mandip is 6m each, what is the distance between Reshma and Mandip?

Answer 1

Hint: Reshma, Salma and Mandip will form a triangle. The radius of circumcircle formed by Reshma, Salma and Mandip is 5m .Length of 2 sides that is distance between Reshma and Salma and between Salma and Mandip are 6m each. By using geometry we can find another side which is the distance between Reshma and Mandip.

Complete step-by-step solution:
In the above figure let’s assume Salma is standing at point S, Reshma is standing at point R and Mandip is standing at point M. H is the mid-point of RM. D is the center of the circle.
SR=SM=6m
Radius DS=DR=DM=5m
We have to find the value of RM.
In triangle SRH applying Pythagoras theorem we get
$R{{H}^{2}}+S{{H}^{2}}=S{{R}^{2}}$
$SR=6m$ , $SH=SD+DH$ and $SD=5m$ applying to the equation
$R{{H}^{2}}+{{\left( 5+DH \right)}^{2}}=36$ …..equation (i)
Again applying Pythagoras theorem to triangle DRH we get
$D{{H}^{2}}+R{{H}^{2}}=R{{D}^{2}}$
$RD=5m$
$D{{H}^{2}}+R{{H}^{2}}=25$ ….equation (ii)
Subtracting equation ii from i
$25+10DH=11$
$\Rightarrow $$DH=-1.4m$
DH is coming out to be negative, which means point D lies outside the triangle. The distance between D and H is 1.4m.
So the actual figure will be -

$\Rightarrow $${{1.4}^{2}}+R{{H}^{2}}=25$
$\Rightarrow $$R{{H}^{2}}=25-1.96=23.04$
$\Rightarrow $$RH=4.8m$
Length of RM is two times of RH
That means $RM=2\times 4.8=9.6m$

Note: Always remember some basic formula in geometry like Pythagoras theorem in order to solve this type of question. When we solve the question by geometry, sometimes the length of line will be a negative value and some students might think that their answer is wrong but that is not always the case like above question. That simply means the location of one end of line segment will change to its opposite side, the length of line will not change.