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The distance formula is a fundamental tool in Analytic Geometry used to determine the distance between two points in space. It is derived from the Pythagorean theorem and is expressed as follows for points in 3-dimensional space:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]
This formula calculates the straight-line distance or the "direct path" between two points, taking into account the differences in all three coordinate dimensions: X, Y, and Z.

• X-coordinate: Differences in horizontal distance.
• Y-coordinate: Differences in vertical distance.
• Z-coordinate: Differences in depth or elevation.

By plugging in the coordinates of the two points into the formula, you can determine the exact distance. This is especially useful in both 2D and 3D geometry when calculating lengths essential for solving geometric problems.

A right triangle is a type of triangle where one of its angles is exactly 90 degrees. This means that one side of the triangle, called the hypotenuse, is opposite the right angle and it's the longest side. The other two sides are referred to as the legs.
The Pythagorean theorem is a key identifier of right triangles, which states:
\[ a^2 + b^2 = c^2 \]
- Here, "a" and "b" are the lengths of the legs,- "c" is the length of the hypotenuse.
In the context of a triangle formed by points like in our problem, if the sum of the squares of two sides equals the square of the third side, the triangle is right-angled. For the given triangle \( PQR \), this relationship holds true, confirming that \( PQR \) is a right triangle.

An isosceles triangle features at least two sides of equal length. This equality in sides gives rise to specific properties:

• Base Angles: The angles opposite the equal sides are also equal.
• Axis of Symmetry: It often possesses a line of symmetry through the vertex angle, bisecting the base.

Identifying an isosceles triangle involves checking for equal side lengths. For our triangle \( PQR \), we computed the following side lengths:

• \( PQ = 3 \)
• \( PR = 6 \)
• \( QR \approx 6.71 \)

Since no two sides have the same length in \( PQR \), it does not meet the criteria for being an isosceles triangle.

3D geometry expands the concepts of basic geometry into three dimensions, allowing for analysis in space using coordinates \((x, y, z)\). This adds depth to traditional 2D figures, permitting analysis beyond flat planes.
In 3D geometry, you will deal with shapes like cubes, spheres, polyhedra, and complex solids. Points are defined by three coordinates, and calculations involve formulas specific to these spatial dimensions.
Key principles include:

• Understanding vectors and their operations.
• Calculating volumes and surface areas of 3D shapes.
• Using coordinate systems to solve problems involving spatial reasoning.

These concepts are crucial for solving geometric problems that occur in our three-dimensional world, providing a framework for real-world applications, such as architectural design, physics, and engineering tasks. The triangle \( PQR \) in the problem above is handled within this framework to calculate distances and determine its properties in three-dimensional space.