Question

A school, hospital, and a supermarket are located at the vertices of a right triangle formed by three highways. The school and hospital are 14.7 miles apart. The distance between the school and the supermarket is 8.82 miles, and the distance between the hospital and the supermarket is 11.76 miles.

A service road will be constructed from the main entrance of the supermarket to the highway that connects the school and hospital. What is the shortest possible length for the service road? Round to the nearest tenth.

Answer 1

7.1 miles

Explanation:

Consider right triangle HospitalSchoolSupermarket. In this triangle:

• HospitalSchool = 14.7 mi;
• HospitalSupermaket = 11.76 mi;
• School Supermarket = 8.82 mi.

The shortest road from the main entrance of the supermarket to the highway that connects the school and hospital will be the height drawn from the point Supermarket to the hypotenuse HospitalSchool.

Let the length of this road be x mi and the distance from School to point A be y mi. Use twice the Pythagorean theorem for right triangles Supermarket SchoolA and SupermarketHospitalA:

`\left\{\begin{array}{l}x^2+y^2=8.82^2\\ \\x^2+(14.7-y)^2=11.76^2\end{array}\right.`

Subtract from the second equation the first one:

`x^2+(14.7-y)^2-x^2-y^2=11.76^2-8.82^2\\ \\14.7^2-2\cdot 14.7y+y^2-y^2=11.76^2-8.82^2\\ \\-29.4y=11.76^2-8.82^2-14.7^2\\ \\29.4y=155.5848\\ \\y\approx5.24\ mi`

Thus,

`x^2=8.82^2-5.24^2=50.3348\\ \\x\approx 7.1\ mi.`