Integrated Math 2 / Unit 7: Right Triangle Trigonometry
7.1 The Pythagorean Theorem
Notes
In a right triangle with legs $a$ and $b$ and hypotenuse $c$: $a^2 + b^2 = c^2$.
- The hypotenuse is the longest side, opposite the right angle.
- Example: legs 9 and 12 give $c^2 = 81 + 144 = 225$, so $c = 15$.
- A Pythagorean triple is a set of whole numbers that fits, such as $3, 4, 5$ and $5, 12, 13$.
- The converse: if $a^2 + b^2 = c^2$ then the triangle is right.
Practice
A right triangle has legs 9 and 12. What is the hypotenuse?
$9^2 + 12^2 = 81 + 144 = 225$ and $\displaystyle \sqrt{225} = 15$.
A 15 ft ladder leans on a wall with its base 9 ft from the wall. How high up the wall does it reach?
The ladder is the hypotenuse: $15^2 - 9^2 = 225 - 81 = 144$, and $\displaystyle \sqrt{144} = 12$.
Which set of side lengths forms a right triangle?
$5^2 + 12^2 = 25 + 144 = 169 = 13^2$. The others fail the test.
Ready for more? Take the Unit 7 test.