The Pythagorean Theorem is a fundamental relationship in geometry that describes the connection between the lengths of the sides of a right triangle. It states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides.
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The Pythagorean Theorem can be expressed mathematically as: $a^2 + b^2 = c^2$, where $a$ and $b$ are the lengths of the two shorter sides, and $c$ is the length of the hypotenuse.
The Pythagorean Theorem is widely used in coordinate geometry to calculate the distance between two points on a coordinate plane.
In right triangle trigonometry, the Pythagorean Theorem is used to find the unknown side lengths of a right triangle given the lengths of the other two sides.
The Pythagorean Theorem can be extended to non-right triangles using the Law of Cosines, which relates the lengths of all three sides of a triangle.
The Pythagorean Theorem is one of the most fundamental and widely used theorems in mathematics, with applications in fields such as physics, engineering, and navigation.
Review Questions
Explain how the Pythagorean Theorem is used in the context of coordinate geometry to calculate the distance between two points.
In the rectangular coordinate system, the Pythagorean Theorem can be used to calculate the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$. The distance formula, which is derived from the Pythagorean Theorem, is given by $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. This formula allows you to find the length of the hypotenuse of the right triangle formed by the two points on the coordinate plane, which represents the distance between them.
Describe how the Pythagorean Theorem is applied in the context of right triangle trigonometry to find unknown side lengths.
In right triangle trigonometry, the Pythagorean Theorem is used to find the lengths of the unknown sides of a right triangle, given the lengths of the other two sides. For example, if the lengths of the two shorter sides, $a$ and $b$, are known, the length of the hypotenuse, $c$, can be calculated using the formula $c^2 = a^2 + b^2$. Conversely, if the length of the hypotenuse and one of the shorter sides are known, the length of the remaining side can be found using the same formula.
Explain how the Pythagorean Theorem is extended to non-right triangles using the Law of Cosines, and describe the relationship between the two theorems.
While the Pythagorean Theorem is specific to right triangles, the Law of Cosines is a more general theorem that can be applied to any triangle, regardless of whether it is a right triangle or not. The Law of Cosines states that in any triangle, the square of the length of one side is equal to the sum of the squares of the lengths of the other two sides, minus twice the product of the lengths of those two sides and the cosine of the angle between them. This relationship reduces to the Pythagorean Theorem when the angle between the two sides is 90 degrees, as the cosine of 90 degrees is 0. Therefore, the Law of Cosines can be seen as a generalization of the Pythagorean Theorem to non-right triangles.
Related terms
Right Triangle: A triangle with one angle measuring 90 degrees, or a square corner.
The longest side of a right triangle, which is opposite the 90-degree angle.
Converse of the Pythagorean Theorem: 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.