Answer:
I come up with n = 10.39. But please review carefully.
Step-by-step explanation:
This is a bit tricky, but I came up with an answer of 10.39 for the measure of n.
1. Develop three equations for the three right triangles represented in the diagram. See the attachment for my labels for the three triangles, plus a measure, p, of the bottom line.
A and B are the two inner triangles, and C is the large triangle with base p, side m, and hypotenuse of (6+18) = 24. Construct an equation for each triangle:
C: m^2 + p^2 = 24^2
B: n^2 + 18^2 = p^2
A: n^2 + 6^2 = m^2
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Use the value of p in equation B and substitute it into equation C:
Equation C:
m^2 + p^2 = 24^2
m^2 + (n^2 + 18^2 ) = 24^2
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Now use the value of m in equation A in the above equation:
m^2 + (n^2 + 18^2 ) = 24^2
(n^2 + 6^2) + (n^2 + 18^2 ) = 24^2
2n^2 +36 + 324 = 576 [Simplify] Both m and p disappear, so we can solve for n, the desired number!
2n^2 = 216
n^2 = 108
n = 10.39
There is probably a way to express this in the format requested, which is a whole number times a square root of another number.
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