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Difficulty: MediumDistance and Midpoint Formulas

In the standard (x,y)(x, y) coordinate plane, where 11 unit represents 11 kilometer, a playground is located at (1,10)(1, 10) and a picnic shelter is located at (5,−2)(5, -2). A water fountain is to be placed exactly halfway between the playground and the picnic shelter. What is the distance, in kilometers, from the origin (0,0)(0, 0) to the water fountain?

Answer: 5 km

Answer

The distance from the origin to the water fountain is 55 kilometers.
The midpoint of the segment representing the water fountain is determined to be (3,4)(3, 4). The distance from the origin (0,0)(0, 0) to (3,4)(3, 4) is found using the Pythagorean relation, yielding 32+42=5\sqrt{3^2 + 4^2} = 5 kilometers.

Step-by-Step Solution

1
Find the coordinates of the midpoint between (1,10)(1, 10) and (5,−2)(5, -2).
The midpoint is (3,4)(3, 4).
The midpoint formula is M=(x1+x22,y1+y22)M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right). Substituting the given endpoints gives M=(1+52,10+(−2)2)=(3,4)M = \left(\frac{1 + 5}{2}, \frac{10 + (-2)}{2}\right) = (3, 4).
2
Calculate the distance from the origin (0,0)(0, 0) to the midpoint (3,4)(3, 4).
The distance is 55 kilometers.
The distance formula from the origin to any point (x,y)(x, y) is d=x2+y2d = \sqrt{x^2 + y^2}. Substituting the coordinates (3,4)(3, 4) yields d=32+42=9+16=25=5d = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5.

Key Concept

Using the midpoint formula to find the center of a line segment, followed by the distance formula from the origin to that point.
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Topics

Distance and Midpoint Formulas

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