Park Project

[BMAD]

I am planning a new community project. I wish to build a simple community park. Currently I am developing access to the park, as the area is rural and this part is really underdeveloped. So we plan to build roads to and have trails through the park. Three straight roads will be used to surround the park while two straight bike trails will connect all three roads together. One of the roads will be 6 miles long; another will be 8 miles long. One bike trail will start at the median point of the 6 mile road and go to the corner of the other two roads. The other bike trail will start at the median point of the 8 mile road and go to the corner of the other two roads. The two trails intersect at a right angle. What is the length of the third road? How big is the park?

[Rob_Gandy]

I'm not quite sure how to start on this one, but from drawing I think that the third road is approx. 4.4 miles long.

[BMAD]

I'm not quite sure how to start on this one, but from drawing I think that the third road is approx. 4.4 miles long.

A number of approaches to this problem are possible. One approach begins by drawing line segment between the medians and considering all of the right-angled triangles in the diagram.

[BobbyGo]

3.535533...?

[BobbyGo]

3.535533...?

Oh wow, I was horribly off. (It would have helped to read the OP a bit more thoroughly)

4.330127... miles

(sqrt(18.75))

[bonanova]

The median lines cut each other into lengths

a and b (which form a right triangle with half of the 6 side)

c and d (which form a right triangle with half of the 8 side.)

a and c form a right triangle with the unknown side x

b and d form a right triangle with a line of length x/2 which joins the 6 and 8 sides at their midpoints.

Thus

a² + b² = 9

c² + d² = 16

a² + c² = x²

b² + d² = x²/4.

Adding the first two equations and subtracting the last equation gives

a² + c² = 9 + 16 - x²/4 which must also equal x² by the third equation. Thus, x² = 20.

x = sqrt (20) = 4.472 miles

By Heron's formula, the area is 13.266 sq miles.

[BMAD]

Can anyone solve the problem with vectors?