Question

The distance between the points $\left( {2,3} \right){\text{ and }}\left( { - 4,5} \right)$ is ____________
\[
  A.{\text{ 2}}\sqrt 2 \\
  B.{\text{ 2}}\sqrt {10} \\
  C.{\text{ 2}}\sqrt {17} \\
  D.{\text{ }}\sqrt {10} \\
 \]

Answer 1

Hint: Use the formula for distance in Coordinate geometry in order to solve the problem. The formula for distance in coordinate geometry is given by $D = \sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}} $.

Complete step-by-step answer:
Given points are $\left( {2,3} \right){\text{ and }}\left( { - 4,5} \right)$
As we know that for any two general points on the Cartesian plane say $A\left( {{x_1},{y_1}} \right){\text{ and }}B\left( {{x_2},{y_2}} \right)$ .
The distance between these points are given by $D = \sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}} $
So, for the above question, we have
\[
  A\left( {{x_1},{y_1}} \right) = \left( {2,3} \right) \\
  B\left( {{x_2},{y_2}} \right) = \left( { - 4,5} \right) \\
 \]
Therefore using the above formula of distance, we get
$
  D = \sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}} \\
   \Rightarrow D = \sqrt {{{\left( { - 4 - 2} \right)}^2} + {{\left( {5 - 3} \right)}^2}} \\
   \Rightarrow D = \sqrt {{{\left( { - 6} \right)}^2} + {{\left( 2 \right)}^2}} \\
   \Rightarrow D = \sqrt {36 + 4} \\
   \Rightarrow D = \sqrt {40} \\
   \Rightarrow D = \sqrt {4 \times 10} = \sqrt {2 \times 2 \times 10} = 2\sqrt {10} \\
 $
Hence, the distance between the points $\left( {2,3} \right){\text{ and }}\left( { - 4,5} \right)$ is $2\sqrt {10} $
So, option B is the correct option.

Note: The distance formula is a useful tool in finding the distance between two points in coordinate geometry. The distance formula itself is actually derived from the Pythagorean Theorem. For solving such types of questions, students must remember the distance formula.