How do I find the values of x & y without using the Pythagorean theorem? It’s okay if you use it to give me the answer however, I don’t understand how you wouldn’t use it.

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  23) x=20; y=10√3

  24) x=4√2; y=8

  25) x=12; y=3+6√3

Step-by-step explanation:

Recognize that all of the triangles involved here are one of the two "special" triangles. By previous use of the Pythagorean theorem, or your familiarity with the trig functions of the "special" angles 30°, 45°, 60°, you know the following side ratios:

  45°-45°-90° triangle: 1 : 1 : √2

  30°-60°-90° triangle: 1 : √3 : 2

Using these ratios, you can solve these figures "by inspection."

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23) x = 2((10√2)/√2) = 20

  y = √3((10√2)/√2) = 10√3

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24) x = 4√2

  y = (4√2)√2 = 8

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25) x = 6·2 = 12

y = 3 +6√3 . . . . . 3 units more than the longer leg of the 6 : 6√3 : 12 triangle