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x, y and z are the 3 unknew, so you need 3 equations.
In the big triangle, by using the Pythagorean's theorem:
y² + z² = (3 + 12)²
y² + z² = 15²
z² = 15² - y² ← memorize this result as (1)
In the little triangle, by using the Pythagorean's theorem:
3² + x² = y²
x² = y² - 3² ← memorize this result as (2)
In the medium triangle, by using the Pythagorean's theorem:
x² + 12² = z² → recall (1): z² = 15² - y²
x² + 12² = 15² - y² → recall (2): x² = y² - 3²
y² - 3² + 12² = 15² - y²
2y² = 15² - 12² + 3²
2y² = 90
y² = 45
→ y = 3√5
Recall (2): x² = y² - 3² = 45 - 9 = 36
→ x = 6
Recall (1): z² = 15² - y² = 15² - 45 = 180
→ z = 6√5
?
z^2 + y^2 = (3+12)^2
z^2 + y^2 = 225
x^2 + 3^2 = y^2
x^2 + 9 = y^2
x^2 + 12^2 = z^2
x^2 + 144 = z^2
x^2 = z^2 - 144
z^2 - 144 + 9 = y^2
z^2 + y^2 = 225
z^2 - 135 = y^2
2z^2 = 360
z^2 = 180
z = 6 root 5
180 + y^2 = 225
y^2 = 45
y = 3 root 5
x^2 = 36
x = 6
xyz = (6 x 3 x 5 x 6) = 540
I'll actually be shocked if I got this right. Honestly it's possible that geometry could have gotten me the answer quicker. But I forget geometry.