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How do you find the distance between the points \[\left( { - 4,2} \right)\], \[\left( {4,17} \right)\]?

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Last updated date: 20th Jun 2024
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Answer
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Hint: Here in this question, we have to find the distance between the two given coordinates. Take the coordinates of two points you want to find the distance between. Call one point. Point 1 \[\left( {{x_1},{y_1}} \right)\] and make the other Point 2 \[\left( {{x_2},{y_2}} \right)\]. Know the distance formula \[\sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}} \] . This formula finds the length of a line that stretches between two points: Point 1 and Point 2.

Complete step-by-step solution:
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. In general the distance can be found by subtracting the coordinates that are not the same.
The distance between two points of the \[xy\] -plane can be found using the distance formula. An ordered pair \[\left( {x,{\text{ }}y} \right)\] 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.
Formula to find Distance Between Two Points in 2d plane. Consider two points, point 1 \[\left( {{x_1},{y_1}} \right)\] and point 2 \[\left( {{x_2},{y_2}} \right)\] on the given coordinate axis.
The distance between these points is given as: \[d = \sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}} \]
Now consider the given two coordinates, point 1 \[\left( { - 4,2} \right)\] and point 2 \[\left( {4,17} \right)\].
The distance between point 1 and point 2 is
\[ \Rightarrow \,\,d = \sqrt {{{\left( {4 - \left( { - 4} \right)} \right)}^2} + {{\left( {17 - 2} \right)}^2}} \]
\[ \Rightarrow \,\,d = \sqrt {{{\left( {4 + 4} \right)}^2} + {{\left( {17 - 2} \right)}^2}} \]
\[ \Rightarrow \,\,d = \sqrt {{{\left( 8 \right)}^2} + {{\left( {15} \right)}^2}} \]
\[ \Rightarrow \,\,d = \sqrt {64 + 225} \]
\[ \Rightarrow \,\,d = \sqrt {289} \]
As we know the 289 is the square number of 17, then
\[ \Rightarrow \,\,d = \sqrt {{{17}^2}} \]
\[\therefore \,\,d = 17\]
Hence, the distance between the points \[\left( { - 4,2} \right)\] and \[\left( {4,17} \right)\] is \[d = 17\].

Note: The distance is a length between the two points. In the geometry we have a formula to determine the distance between the points. While determining the distance between the points we consider the both values of x and the value of y. Where x and y are the coordinates.