Find the distance between the pair of points (2,3) and (8,9). If necessary, express the answer in simplified radical form and then round to two decimal places.
Answer (1)
Apply the distance formula: d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 .
Substitute the coordinates (2,3) and (8,9) into the formula: d = ( 8 − 2 ) 2 + ( 9 − 3 ) 2 .
Simplify the expression: d = 6 2 + 6 2 = 72 = 6 2 .
Round to two decimal places: 6 2 ≈ 8.49 .
6 2
Explanation
State the distance formula and given points. We are given two points (2,3) and (8,9) and asked to find the distance between them. We will use the distance formula to find the distance. The distance formula is given by:
d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2
where ( x 1 , y 1 ) and ( x 2 , y 2 ) are the coordinates of the two points.
Substitute the coordinates. Substitute the given coordinates into the distance formula:
d = ( 8 − 2 ) 2 + ( 9 − 3 ) 2
Simplify the expression. Simplify the expression inside the square root:
d = ( 6 ) 2 + ( 6 ) 2 d = 36 + 36 d = 72
Simplify the radical. Simplify the radical:
72 = 36 ⋅ 2 = 6 2
Round to two decimal places. Round the result to two decimal places. We know that 72 ≈ 8.48528137423857 . Rounding to two decimal places, we get 8.49.
State the final answer. The distance between the points (2,3) and (8,9) is 6 2 units, which is approximately 8.49 units.
Examples
The distance formula is used in navigation to calculate the shortest distance between two points on a map or a screen. For example, a GPS device uses coordinates to locate your position and the position of your destination. The device then uses the distance formula to calculate the distance between your current location and your destination, providing you with an estimate of how far you need to travel. This is also applicable in fields like logistics and urban planning to optimize routes and infrastructure.
