Geometry / Unit 9: Right Triangles and Trigonometry
9.1 The Pythagorean Theorem
Notes
In a right triangle, the legs form the right angle and the hypotenuse is the longest side, opposite the right angle.
- Pythagorean Theorem: $a^2 + b^2 = c^2$, where $c$ is the hypotenuse.
- To find a leg: $a^2 = c^2 - b^2$.
- Pythagorean triples are whole-number solutions: 3-4-5, 5-12-13, 8-15-17, 7-24-25, and their multiples.
- Example: legs 6 and 8 give $\displaystyle c = \sqrt{36 + 64} = 10$.
Practice
A right triangle has legs 9 and 40. Find the hypotenuse.
$\displaystyle \sqrt{81 + 1600} = \sqrt{1681} = 41$.
A right triangle has a hypotenuse of 25 and one leg of 7. Find the other leg.
$\displaystyle \sqrt{625 - 49} = \sqrt{576} = 24$.
A 13-foot ladder leans against a wall. Its base is 5 feet from the wall. How high up the wall does it reach?
$\displaystyle \sqrt{169 - 25} = \sqrt{144} = 12$.
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