Answer:
9. (x, y) = (6√3, 3)
10. (x, y) = (14, 14√2)
11. (x, y) = (2√6, 3√2)
12. (x, y) = (6, 2)
Explanation:
Because you have memorized a short table of trig functions, you know that the ratio of side lengths of a 30°-60°-90° triangle is 1 : √3 : 2, and the ratio of side lengths of a 45°-45°-90° triangle is 1 : 1 : √2.
In each case, we compare the given side ratios to the known ratios for the kind of triangle we have. Then we multiply the triangle ratios by a scale factor that makes the given number match the corresponding ratio value. Matching the other ratio values, we can determine the values of the variables.
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9. Using the side ratios for the 30-60-90 triangle, you have
6 : x : y+9 = 1 : √3 : 2
Multiplied by 6, the ratios on the right are ...
6 : x : y+9 = 6 : 6√3 : 12
x = 6√3
y +9 = 12
y = 3
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10. Using the side ratios for the 45-45-90 triangle:
14 : x : y = 1 : 1 : √2
Multiplying the ratios on the right by 14, we have ...
14 : x : y = 14 : 14 : 14√2
x = 14
y = 14√2
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11. Again using the 30-60-90 ratios:
√6 : y : x = 1 : √3 : 2
Multiplying the ratios on the right by √6, we have ...
√6 : y : x = √6 : 3√2 : 2√6
y = 3√2
x = 2√6
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12. Again, using the 45-45-90 ratios:
x : 3y : 6√2 = 1 : 1 : √2
Multiplying the ratios on the right by 6, we have ...
x : 3y : 6√2 = 6 : 6 : 6√2
x = 6
3y = 6
y = 2