Question

Consider the triangle with vertices \(P(2,0,-3), Q(5,-2,1),\) and \(R(7,5,3)\) a. Show that it is a right-angled triangle. b. Find the lengths of the three sides and verify the Pythagorean theorem.

Short answer

The triangle is right-angled at \( Q \), and the side lengths satisfy the Pythagorean theorem.

Step-by-step solution

Find Vectors of the Sides

To find the vectors representing the sides of the triangle, calculate the differences of the coordinates of the vertices.\\( \overrightarrow{PQ} = (5-2, -2-0, 1-(-3)) = (3, -2, 4) \)\\( \overrightarrow{QR} = (7-5, 5-(-2), 3-1) = (2, 7, 2) \)\\( \overrightarrow{RP} = (2-7, 0-5, -3-3) = (-5, -5, -6) \)

Verify Right Angle with Dot Product

To show the triangle is right-angled, check if any two sides are perpendicular by calculating the dot product of their vectors.\Calculate \( \overrightarrow{PQ} \cdot \overrightarrow{QR} \):\\[ 3 \times 2 + (-2) \times 7 + 4 \times 2 = 6 - 14 + 8 = 0 \] \Since the dot product is zero, \( \overrightarrow{PQ} \) and \( \overrightarrow{QR} \) are perpendicular, indicating a right angle at point \( Q \).

Calculate Lengths of the Sides

Use the distance formula to find the lengths of the sides. \\( PQ = \sqrt{3^2 + (-2)^2 + 4^2} = \sqrt{9 + 4 + 16} = \sqrt{29} \) \\( QR = \sqrt{2^2 + 7^2 + 2^2} = \sqrt{4 + 49 + 4} = \sqrt{57} \)\\( RP = \sqrt{(-5)^2 + (-5)^2 + (-6)^2} = \sqrt{25 + 25 + 36} = \sqrt{86} \)

Verify Pythagorean Theorem

To confirm the triangle is right-angled, verify the Pythagorean theorem: \( PQ^2 + QR^2 = RP^2 \).Check:\[ (\sqrt{29})^2 + (\sqrt{57})^2 = \sqrt{86})^2 \]\[ 29 + 57 = 86 \]Since the equation holds, the sides satisfy the Pythagorean theorem.

Key concepts

Vector Calculation

Vectors are fundamental in understanding the relationships between points in space. They tell us about direction and magnitude. When dealing with a triangle formed by points in 3D space, each side of the triangle can be represented as a vector. To form this vector, you subtract the coordinates of the initial point from the coordinates of the terminal point.

For example, if you have two points, \( A(x_1, y_1, z_1) \) and \( B(x_2, y_2, z_2) \), the vector \( \overrightarrow{AB} \) is given by:

• \( \overrightarrow{AB} = (x_2 - x_1, y_2 - y_1, z_2 - z_1) \)

This gives us a way to compute vectors for each side of the triangle by using the vertex coordinates.

Dot Product

The dot product, in simple terms, is a way to find the relationship between two vectors. Specifically, it helps in determining if two vectors are perpendicular. The dot product \( \overrightarrow{A} \cdot \overrightarrow{B} \) between two vectors \( \overrightarrow{A} = (a_1, a_2, a_3) \) and \( \overrightarrow{B} = (b_1, b_2, b_3) \) is calculated as:

\[ a_1b_1 + a_2b_2 + a_3b_3 \]

If the result is zero, the vectors are perpendicular, which indicates a right angle in geometry. This property is crucial to verify if a triangle is right-angled by checking the dot product of vectors along its sides.

Distance Formula

The distance formula helps us find the length of a side represented by a vector in 3D space. This is essentially the Euclidean distance derived from the Pythagorean theorem for 3D coordinates. The distance between two points \( P(x_1, y_1, z_1) \) and \( Q(x_2, y_2, z_2) \) is calculated as:

• \( \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \)

This formula is used to find the lengths of all sides of the triangle. It helps determine the measure of each edge, which is essential for verifying properties such as the Pythagorean theorem.

Pythagorean Theorem

The Pythagorean theorem is a vital concept often used to identify right-angled triangles. For a triangle with sides \( a \), \( b \), and \( c \) (where \( c \) is the hypotenuse), the theorem states that:

\[ a^2 + b^2 = c^2 \]

This relationship shows that the sum of the squares of the two shorter sides is equal to the square of the longest side. To verify if a triangle is right-angled, you calculate the squares of its side lengths and see if this equation holds true. Using the lengths from the distance formula, you can easily apply this theorem.

Coordinate Geometry

Coordinate geometry provides a systematic way to analyze geometric shapes using a coordinate system. In the context of the right-angled triangle problem, it aids in simplifying the process of calculating vectors, distances, and verifying angles.

The coordinates of points give a clear structure that allows for computational geometry, which involves using algebra to solve geometric problems. Coordinate geometry bridges the gap between algebra and geometry, making it easier to handle complex problems like finding the type of triangle and its properties using calculations directly derived from point coordinates.