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Find the distance between (2, 3) and (-4, 5).
A. $2\sqrt 2 $
B. $2\sqrt {10} $
C. $2\sqrt {17} $
D. $\sqrt {10} $

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Last updated date: 18th Jun 2024
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Answer
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Hint: We will use the distance formula $d = \sqrt {{{({x_2} - {x_1})}^2} + {{({y_2} - {y_1})}^2}} $ where the two points are $({x_1},{y_1})$ and $({x_2},{y_2})$ and thus equated with the given four options and thus, we will get our answer.

Complete step-by-step answer:
Let us first discuss the distance formula:-
Let the two points be $({x_1},{y_1})$ and $({x_2},{y_2})$. So, the distance d between between the given two points are given by: $d = \sqrt {{{({x_2} - {x_1})}^2} + {{({y_2} - {y_1})}^2}} $. This distance is known as Euclidean distance. This is an application of Pythagorean Theorem.
The Pythagorean Theorem consists of a formula ${a^2} + {b^2} = {c^2}$ which is used to figure out the value of (mostly) the hypotenuse in a right triangle. The a and b are the two "non-hypotenuse" sides of the triangle (Opposite and Adjacent).
So, if we compare the two given points (2, 3) and (-4, 5) with $({x_1},{y_1})$ and $({x_2},{y_2})$, we will have:-
${x_1} = 2$, ${y_1} = 3$, ${x_2} = - 4$ and ${y_2} = 5$.
Putting all these values in the mentioned distance formula that is $d = \sqrt {{{({x_2} - {x_1})}^2} + {{({y_2} - {y_1})}^2}} $.
So, we will get the following expression:
$d = \sqrt {{{( - 4 - 2)}^2} + {{(5 - 3)}^2}} $
$\therefore d = \sqrt {{{( - 6)}^2} + {{(2)}^2}} $
$\therefore d = \sqrt {36 + 4} $
$\therefore d = \sqrt {40} $
There is no such option.
So, we will simplify the number inside the root more to get the desired value.
$40 = 2 \times 2 \times 2 \times 5$
Therefore, $\sqrt {40} = \sqrt {2 \times 2 \times 2 \times 5} = 2 \times \sqrt {2 \times 5} $.
Hence, $d = 2\sqrt {10} $.

So, the correct answer is “Option B”.

Note: Students must note that even if you interchange the points in the distance formula, it would not affect the result which says that if two points are A and B, then AB = BA. This is happening because we are eventually squaring the difference of terms which will neglect – sign.
Sometimes, it is also possible that you may have the option of without simplified value inside the root and after simplifying, we would not be able to match it to any option. So, always remember to check the earlier result before you simplify.
Fun Fact:- There are a lot of types of differences in real numbers only. There is even one which is known as Taxi-Cab distance.