The Pythagorean Theorem
Apply the Pythagorean Theorem a^2+b^2=c^2 to find missing sides of right triangles: use it to verify right angles, calculate distances, and solve real-world geometry problems.
Key Concepts
Property If a triangle is a right triangle, the sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse. For legs $a$ and $b$, and hypotenuse $c$, $a^2 + b^2 = c^2$.
A right triangle has a leg of 9 cm and a hypotenuse of 15 cm. Find the other leg, $b$: $9^2 + b^2 = 15^2 \rightarrow 81 + b^2 = 225 \rightarrow b^2 = 144 \rightarrow b = 12$ cm. A right triangle has legs of 7 in and 24 in. Find the hypotenuse, $c$: $7^2 + 24^2 = c^2 \rightarrow 49 + 576 = c^2 \rightarrow 625 = c^2 \rightarrow c = 25$ in.
Think of this as a magic recipe for right triangles! If you know the lengths of the two shorter sides (the legs), you can find the length of the longest side, the hypotenuse. Just square the legs, add them up, and the result is the square of the hypotenuse. It's a trusty shortcut for finding any missing side!
Common Questions
What does the Pythagorean Theorem state and what are its variables?
The Pythagorean Theorem states that in a right triangle, a^2+b^2=c^2, where a and b are the legs (the two shorter sides) and c is the hypotenuse (the side opposite the right angle). This relationship holds for every right triangle.
How do you find a missing leg when the hypotenuse and one leg are known?
Substitute the known values into a^2+b^2=c^2. Solve for the unknown leg. For example, if the hypotenuse is 13 and one leg is 5: 5^2+b^2=13^2 gives 25+b^2=169, so b^2=144 and b=12.
How is the Pythagorean Theorem used to find distances on the coordinate plane?
The distance formula is derived from the Pythagorean Theorem. The horizontal distance between two points is the horizontal leg, the vertical distance is the vertical leg, and the straight-line distance is the hypotenuse. d=sqrt((x2-x1)^2+(y2-y1)^2) is the Pythagorean Theorem applied to coordinate geometry.