Question

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

Answer 1

Hint:
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 \[\left( {x,y} \right)\].