1. Given a right triangle with legs $a=12$, $b=5$, and hypotenuse $c$, we use the Pythagorean theorem: $$c^2 = a^2 + b^2$$ Substitute values: $$c^2 = 12^2 + 5^2 = 144 + 25 = 169$$ Take the square root: $$c = \sqrt{169} = 13$$ 2. Given $a=8$, $c=10$, find $b$: Using the Pythagorean theorem: $$b^2 = c^2 - a^2 = 10^2 - 8^2 = 100 - 64 = 36$$ Take the square root: $$b = \sqrt{36} = 6$$ 3. Given $b=30$, $c=34$, find $a$: Using Pythagorean theorem: $$a^2 = c^2 - b^2 = 34^2 - 30^2 = 1156 - 900 = 256$$ Take the square root: $$a = \sqrt{256} = 16$$