Question

What is the distance between the origin and the point $\left( { - 19,6} \right)$ ?

Answer 1

Hint: In the given question, we are required to find the distance of the point $\left( { - 19,6} \right)$ from the points origin on the Cartesian plane. We will use the distance formula to solve it.

Formula used:
Distance between the points $(x_1,y_1)$ and $(x_2,y_2)$ is given as $d = \sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}} $
Where, ${x_2} = $ x-coordinate of second point
${x_1} = $ x-coordinate of first point
${y_2} = $ y-coordinate of second point
${y_1} = $ y-coordinate of first point
By using this formula, we can find the distance between any given two points.

Complete step by step answer:
Let us name the given points as $P\left( { - 19,6} \right)$ . Now, we have to find the distance of the point P whose coordinates are $\left( { - 19,6} \right)$ from the origin on a Cartesian plane.
Now, we should know that the coordinates of the origin on the Cartesian plane are given by: $\left( {0,0} \right)$. Hence, we can calculate the distance of the point P with coordinates $\left( { - 19,6} \right)$ from the origin with coordinates as $\left( {0,0} \right)$ using the distance formula.
So, we have, ${x_1} = - 19$, ${x_2} = 0$, ${y_1} = 6$ and ${y_2} = 0$.
Now, substituting the values of the known quantities into the distance formula $d = \sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}} $, we get,
$ \Rightarrow d = \sqrt {{{\left( {0 - \left( { - 19} \right)} \right)}^2} + {{\left( {0 - 6} \right)}^2}} $
Simplifying the expression, we get,
$ \Rightarrow d = \sqrt {{{\left( {19} \right)}^2} + {{\left( { - 6} \right)}^2}} $
Computing the squares, we get,
$ \Rightarrow d = \sqrt {361 + 36} $
Adding up the like terms, we get,
$ \Rightarrow d = \sqrt {397} $
Now, $397$ is not a perfect square since the factorisation is $397 = 397 \times 1$.
Hence, the distance between the given point $\left( { - 19,6} \right)$ and origin is $\sqrt {397} $ units.

Note:
Distance formula can be used to find the distance between any two given points on the Cartesian plane. The distance between a point and a line can also be calculated using the distance formula if we know a point lying on the given line. We can change the order of ${x_1},{x_2},{y_1}$ and ${y_2}$ as there are square terms inside the square root.