Question

What is the distance of the point $\left( {3,4} \right)$ from the origin?

Answer 1

Here, we need to find the distance of the given point from the origin. Let the point be $P\left( {3,4} \right)$. The origin is the point $O\left( {0,0} \right)$. We will use the distance formula to find the length of the line segment $PO$, and thus, the distance of the point $P\left( {3,4} \right)$ from the origin.

Formula Used:
Distance formula: The distance $d$ between two points $\left( {{x_1},{y_1}} \right)$ and $\left( {{x_2},{y_2}} \right)$ is given by the formula $d = \sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}}$.

Complete step by step solution:
Let the given point be $P\left( {3,4} \right)$.
The origin is the point $\left( {0,0} \right)$.
Let the origin be $O\left( {0,0} \right)$.
We need to find the distance between the points $P$ and $O$, that is the length of the line segment $PO$.
We will use the distance formula to find the distance between the points $O\left( {0,0} \right)$ and $P\left( {3,4} \right)$.
The distance formula states that the distance $d$ between two points $\left( {{x_1},{y_1}} \right)$ and $\left( {{x_2},{y_2}} \right)$ is given by the formula $d = \sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}}$.
Comparing the point $O\left( {0,0} \right)$ to $\left( {{x_1},{y_1}} \right)$, we get
${x_1} = 0$ and ${y_1} = 0$
Comparing the point $P\left( {3,4} \right)$ to $\left( {{x_2},{y_2}} \right)$, we get
${x_2} = 3$ and ${y_2} = 4$
Now, substituting ${x_1} = 0$, ${y_1} = 0$, ${x_2} = 3$, and ${y_2} = 4$ in the distance formula, we get
$\Rightarrow PO = \sqrt {{{\left( {3 - 0} \right)}^2} + {{\left( {4 - 0} \right)}^2}}$
Subtracting the like terms in the parentheses, we get
$\Rightarrow PO = \sqrt {{3^2} + {4^2}}$
Simplifying the expression by applying the exponents on the bases, we get
$\Rightarrow PO = \sqrt {9 + 16}$
Adding the terms in the expression, we get
$\Rightarrow PO = \sqrt {25}$
We know that 25 is the square of 5.
Therefore, rewriting the expression, we get
$\Rightarrow PO = \sqrt {{5^2}}$
Simplifying the expression, we get
$\therefore PO = 5$
Therefore, we get the length of line $PO$ as 5 units.

Thus, we get the distance between the points $P\left( {3,4} \right)$ and $O\left( {0,0} \right)$ as 5 units.

Note:
A common mistake is to write the distance between the points $P\left( {3,4} \right)$ and $O\left( {0,0} \right)$ as $\sqrt {{{\left( {4 - 3} \right)}^2} + {{\left( {0 - 0} \right)}^2}}$. This is incorrect since the distance formula requires the subtraction of the abscissa of the two points, and the ordinate of the two points. In any point of the form $\left( {x,y} \right)$ lying on the cartesian plane, $x$ is called the abscissa of the point $\left( {x,y} \right)$, and $y$ is called the ordinate of the point $\left( {x,y} \right)$.