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Answer:
11. x = 13, y = 13√2
12. x = 8, y = 4√3
13. x = 14√3, y = 7√3
14. x = y = 11√2
15. x = 12√2, y = 12, z = 12√3
Explanation:
The ratios of side lengths in the "special" triangles are (shortest to longest)
1 : 1 : √2 . . . . . isosceles right triangle
1 : √3 : 2 . . . . 30°-60°-90° triangle
These ratios make answering these questions much easier.
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11. 13 : x : y = 1 : 1 : √2
x = 13, y = 13√2
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12. 4 : y : x = 1 : √3 : 2
x = 8, y = 4√3
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13. y : 21 : x = 1 : √3 : 2
y = 21/√3 = 7√3
x = 14√3
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14. x : y : 22 = 1 : 1 : √2
x = 22/√2 = 11√2
x = y = 11√2
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15. Let 'a' represent the altitude. Then we have ...
a : z : 24 = 1 : √3 : 2
a = 12, z = 12√3
and ...
a : y : x = 1 : 1 : √2
y = 12, x = 12√2
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Additional comment
It is worth remembering that 1/√n = (√n)/n. This is useful when you have to divide by √2 or √3 in the above problems.