Question

How do you find the distance between $\left( {43, - 15} \right)$ and $\left( {29, - 3} \right)$?

Answer 1

Hint: According to the given question, we have to find the distance between$\left( {43, - 15} \right)$ and $\left( {29, - 3} \right)$.
So, first of all we have to understand about the concept to find the distance between two points which is mentioned below,
The distance between two points by using the distance formula, this is an application of the Pythagorean Theorem. So, we can rewrite the Pythagorean Theorem as mentioned below,
$ \Rightarrow $ Distance between two points $\left( d \right) = \sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}} ..........................(A)$

Complete step by step solution:
Step 1: First of all we have to understand that $\left( {{x_1},{x_2}} \right)$ is $\left( {43, - 15} \right)$ and $\left( {{y_1},{y_2}} \right)$ is $\left( {29, - 3} \right)$.
Step 2: Now, we have to use the formula (A) for finding the distance between two points,
\[ \Rightarrow \left( d \right) = \sqrt {{{\left( {29 - 43} \right)}^2} + {{\left( { - 3 - \left( { - 15} \right)} \right)}^2}} \]
Now, we have to solve the above expression by adding or subtracting the term,
\[ \Rightarrow \left( d \right) = \sqrt {{{\left( { - 14} \right)}^2} + {{\left( {12} \right)}^2}} \]
Step 3: Now, we have to solve the above expression which is obtained in the solution step 2 by putting the values of${\left( {14} \right)^2}$ and ${\left( {12} \right)^2}$
Now, we have to solve the above expression by adding the terms,
\[ \Rightarrow \left( d \right) = \sqrt {340} \]
Step 4: Now, we have to make the factors of $340$ by the prime factorisation method,
\[ \Rightarrow \left( d \right) = \sqrt {2 \times 2 \times 5 \times 17} \]
Now, we have to take 2 outside from the square root,
\[ \Rightarrow \left( d \right) = 2\sqrt {85} \]
Now, we have to put the square root of $85$ as $9.21$ in the above expression,
\[ \Rightarrow \left( d \right) = 2 \times 9.21\]
Now, we have to solve the above expression by multiplying the terms,
\[ \Rightarrow \left( d \right) = 18.42\]

Final solution: hence, the distance between $\left( {43, - 15} \right)$ and $\left( {29, - 3} \right)$ is \[\left( d \right) = 18.42\].

Note:
1. It is necessary to understand the concept to find the distance between two points which is mentioned in the solution hint.
2. It is necessary to use the formula (A) to find the distance between the points $\left( {43, - 15} \right)$ and $\left( {29, - 3} \right)$.