An airport, a factory, and a shopping center are at the vertices of a right triangle formed by three highways. The airport and factory are 6.0 miles apart. Their distances from the shopping center are 3.6 miles and 4.8 miles, respectively. A service road will be constructed from the shopping center to the highway that connects the airport and factory. What is the shortest possible length for the service road?

Expert Answers

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Check that this triangle is really a right one, for this its sides must satisfy the Pythagorean theorem:

`3.6^2+4.8^2=12.96+23.04=36=6^2.`

Yes, it is right.

The shortest distance from a point to a straight line is along the line perpendicular to the original line. The corresponding line segment is called a height of a triangle.

The simplest way to find length of this height is to consider area of the triangle. It is the half of the product of one side length and the corresponding height, `A=1/2 a h.`

If we take the longest side, hypotenuse, as a "side", we obtain `A=1/2 *6.0*H,` where `H` is the distance in question. But if we consider a shorter side, a cathetus, then the corresponding height is the another cathetus. So `A=1/2 *3.6*4.8.`

The area is the same, so `6.0*H=3.6*4.8,` or `H=(3.6*4.8)/6.0=2.88` (miles). This is the answer.

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