Find the lengths of the sides of the triangle with the...

Find the lengths of the sides of the triangle with the indicated vertices, and determine whether the triangle is a right triangle, an isosceles triangle, or neither.
$$(-1,0,-2),(-1,5,2),(-3,-1,1)$$

Key Concepts

Distance Formula in Three-Dimensional Space

The distance formula in 3D is used to calculate the length of a line segment between two points with coordinates (x1, y1, z1) and (x2, y2, z2). It generalizes the Pythagorean theorem to three dimensions by taking the square root of the sum of the squared differences in each coordinate. This concept is essential when working with triangles defined by vertices in space to obtain the side lengths accurately.

Triangle Classification Using Side Lengths

After computing the side lengths of a triangle, these values can be used to classify the triangle. Classification can be based on the equality of side lengths or the relationship among them. For example, if two sides are equal in length, the triangle is isosceles, whereas unique relationships among the sides may indicate other types of triangles. This concept helps in understanding and determining the nature of the triangle based solely on its geometric dimensions.

Pythagorean Theorem in the Context of Right Triangles

The Pythagorean theorem is a fundamental tool in determining whether a triangle is right-angled. By checking if the square of one side is equal to the sum of the squares of the other two sides, one can verify the presence of a right angle. This application is crucial not only in two dimensions but also when dealing with triangles in three-dimensional space, provided the sides are calculated correctly.

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