BrainDen.com - Brain Teasers

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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.

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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.

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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.

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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))

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