Pythagorean Theorem
Apply the Pythagorean Theorem in Grade 9 math: use a²+b²=c² to find missing side lengths of right triangles, identify hypotenuse versus legs, and solve real-world distance problems.
Key Concepts
Property If a triangle is a right triangle with legs of lengths $a$ and $b$ and hypotenuse of length $c$, then $$a^2 + b^2 = c^2.$$.
Explanation This famous theorem is your superpower for right triangles. It reveals a hidden connection: the area of the squares on the two shorter legs perfectly adds up to the area of the square on the hypotenuse. You can use this balanced equation to find any missing side length, as long as you know the other two sides.
Examples Find the hypotenuse $c$ for a right triangle with legs $a=5$ and $b=12$: $5^2 + 12^2 = c^2 \implies 25 + 144 = c^2 \implies 169 = c^2 \implies c = 13$. Find the leg $b$ for a right triangle with leg $a=8$ and hypotenuse $c=10$: $8^2 + b^2 = 10^2 \implies 64 + b^2 = 100 \implies b^2 = 36 \implies b = 6$. Find the hypotenuse $m$ for legs of 4 and 6: $4^2 + 6^2 = m^2 \implies 16 + 36 = m^2 \implies m = \sqrt{52} = 2\sqrt{13}$.
Common Questions
What does the Pythagorean Theorem state?
The Pythagorean Theorem states that in a right triangle, the square of the hypotenuse (c) equals the sum of the squares of the two legs (a and b): a² + b² = c². This only applies to right triangles.
How do you find a missing leg using the Pythagorean Theorem?
If the hypotenuse is c = 13 and one leg is a = 5, substitute into a² + b² = c²: 25 + b² = 169. Subtract 25: b² = 144. Take the square root: b = 12.
What is the hypotenuse and how do you identify it?
The hypotenuse is the longest side of a right triangle, always opposite the right angle. It is the c term in a² + b² = c², and you can identify it as the side not forming the right angle.