Use the distance formula and the converse of the Pythagorean...

Use the distance formula and the converse of the Pythagorean theorem to show that the points $\mathbf{S}(3,5,2), \mathbf{T}(2,3,-1)$, and $\mathbf{U}(6,1,-1)$ are the vertices of a right triangle.

Key Concepts

Distance Formula

The distance formula allows you to calculate the distance between two points in Euclidean space by taking the square root of the sum of the squares of the differences between corresponding coordinates. In three-dimensional space, given two points (x1, y1, z1) and (x2, y2, z2), the distance is computed as ?[(x2 - x1)² + (y2 - y1)² + (z2 - z1)²]. This formula is fundamental for determining the lengths of sides when dealing with geometrical figures in coordinate space.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry that relates the lengths of the sides of a right triangle. Specifically, it states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. This theorem is commonly used to verify relationships between side lengths in triangles.

Converse of the Pythagorean Theorem

The converse of the Pythagorean theorem is used to determine whether a given triangle is right-angled. It states that if the square of one side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle. This concept is valuable in problems where the side lengths are computed using the distance formula and you need to confirm the presence of a right angle.

Right Triangle

A right triangle is a type of triangle that includes a 90-degree angle. The identification of a right triangle has important consequences in geometry and trigonometry because it allows for the application of the Pythagorean theorem and trigonometric ratios to solve various problems. Recognizing that a triangle is right-angled is often an essential step in analyzing the properties of the triangle.

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