4.3.1 Pythagorean Theorem

Understanding and applying the Pythagorean theorem (a² + b² = c²) to find missing sides in right triangles.

定义

The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the lengths of the other two sides (called legs). If a right triangle has legs of length \(a\) and \(b\), and hypotenuse of length \(c\), then the relationship \(a^2 + b^2 = c^2\) holds. This fundamental theorem is used to find missing side lengths in right triangles and serves as the foundation for trigonometric relationships. The converse is also true: if three positive numbers \(a\), \(b\), and \(c\) satisfy \(a^2 + b^2 = c^2\), then they can form the sides of a right triangle.

核心公式

• \(a^2 + b^2 = c^2\)
• \(c = \sqrt{a^2 + b^2}\)
• \(a = \sqrt{c^2 - b^2}\)
• \(b = \sqrt{c^2 - a^2}\)
• \(\text{Distance formula: } d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)

易错点

• ⚠️ Confusing which side is the hypotenuse: students often apply the theorem to non-right triangles or incorrectly identify the hypotenuse as a leg, leading to wrong calculations. Remember that the hypotenuse is always the longest side and is opposite the right angle.
• ⚠️ Forgetting to take the square root when solving for a missing side: students correctly set up the equation but forget the final step of taking the square root, leaving the answer as a squared value instead of the actual length.
• ⚠️ Incorrectly applying the theorem when given a right triangle in a coordinate plane: students fail to recognize that the distance formula is derived from the Pythagorean theorem and may not properly set up the coordinate differences.
• ⚠️ Misidentifying the right angle in a diagram: students may assume the right angle is at a different vertex than indicated, causing them to use the wrong sides in their calculations.