Question

Given the points \[A\left( {0,0} \right)\] and \[B\left( {6,8} \right)\], how do you find the distance?

Answer 1

Hint: Distance between the points \[A\left( {{x_1},y_1^{}} \right)\] and \[B\left( {{x_2},{y_2}} \right)\] is given by
\[d = \sqrt {{{(x_2^{} - x_1^{})}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}} \]
Here, \[d\] is the distance between the two given points. This formula can be derived using the Pythagoras theorem.

Complete step-by-step solution:
Using the distance formula we can write the following
\[d = \sqrt {{{(x_2^{} - x_1^{})}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}} \]
Substituting the values of the points in the above formula we will get
\[ \Rightarrow d = \sqrt {{{(6 - 0)}^2} + {{\left( {8 - 0} \right)}^2}} \]
On simplifying,
\[ \Rightarrow d = \sqrt {{{(6)}^2} + {{\left( 8 \right)}^2}} \]
Squaring of the numbers under the square root
\[ \Rightarrow d = \sqrt {36 + 64} \]
Adding the numbers under the square root
\[ \Rightarrow d = \sqrt {100} \]
Square root of the above number
\[ \Rightarrow d = 10\]

Therefore, the distance between these two points is \[10\] units.

Note: The distance between two points is the length of the interval joining the two points.
If the two points lie on the same horizontal or same vertical line, the distance can be found by subtracting the coordinates that are not the same.
In analytic geometry, distance formula is used to find the distance measure between two lines, the sum of the lengths of all the sides of a polygon, perimeter of polygons on a coordinate plane, the area of polygons and many more. For example, we can find the lengths of sides of a triangle using the distance formula and determine whether the triangle is scalene, isosceles or equilateral.
The distance between two points of the xy-plane can be found using the distance formula. An ordered pair \[\left( {x,y} \right)\] represents co-ordinate of the point, where x-coordinate (or abscissa) is the distance of the point from the centre and y-coordinate (or ordinate) is the distance of the point from the centre.
Distance between two points in polar coordinates.
Let O be the pole and OX be the initial line. Let P and Q be two given points whose polar coordinates are \[\left( {{r_1},{\theta _1}} \right)\] and \[\left( {{r_2},{\theta _2}} \right)\] respectively. Then the distance formula between the points is given by,
\[PQ = \sqrt {r_1^2 + r_2^2 - 2{r_1}{r_2}\cos \left( {{\theta _1} - {\theta _2}} \right)} \]
where \[{\theta _1}\] and \[{\theta _2}\] in radians.