📚 Topic Summary
The Pythagorean Theorem states that for a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. Mathematically, this is expressed as $a^2 + b^2 = c^2$, where $c$ is the hypotenuse. The Pythagorean Converse is the flip side of this theorem. It states that if the square of the length of the longest side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle.
In simpler terms, if you have a triangle with sides $a$, $b$, and $c$ (where $c$ is the longest side), and $a^2 + b^2 = c^2$, then you know it's a right triangle. This is a super handy tool for determining if a triangle is right-angled without needing to measure the angles!
🧮 Part A: Vocabulary
Match the term with its definition:
| Term |
Definition |
| 1. Hypotenuse |
A. The theorem stating $a^2 + b^2 = c^2$ for a right triangle. |
| 2. Pythagorean Theorem |
B. The side opposite the right angle in a right triangle. |
| 3. Leg |
C. If $a^2 + b^2 = c^2$, then the triangle is a right triangle. |
| 4. Pythagorean Converse |
D. Either of the two sides that form the right angle in a right triangle. |
| 5. Right Triangle |
E. A triangle that has one 90-degree angle. |
✏️ Part B: Fill in the Blanks
Complete the paragraph using the words: converse, right, squares, hypotenuse, theorem.
The Pythagorean _______ states that in a _______ triangle, the square of the _______ is equal to the sum of the _______ of the other two sides. The Pythagorean _______ states that if $a^2 + b^2 = c^2$, then the triangle is a right triangle.
🤔 Part C: Critical Thinking
Can a triangle with sides 5, 12, and 14 be a right triangle? Show your work and explain why or why not.