Question

Determine whether \(\triangle R S T\) with vertices \(R(1,5), S(-1,1)\), and \(T(5,4)\) is scalene. Explain.

Short answer

The triangle \(\triangle RST\) is scalene as all sides are different lengths.

Step-by-step solution

Find Distance RS

Use the distance formula to find the length of side \(RS\). The distance formula is \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). Substituting the coordinates of \(R(1,5)\) and \(S(-1,1)\), we have: \[RS = \sqrt{(-1 - 1)^2 + (1 - 5)^2} = \sqrt{(-2)^2 + (-4)^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5}.\]

Find Distance ST

Use the distance formula to find the length of side \(ST\) using \(S(-1,1)\) and \(T(5,4)\), so: \[ST = \sqrt{(5 - (-1))^2 + (4 - 1)^2} = \sqrt{(6)^2 + (3)^2} = \sqrt{36 + 9} = \sqrt{45} = 3\sqrt{5}.\]

Find Distance RT

Use the distance formula to find the length of side \(RT\) using \(R(1,5)\) and \(T(5,4)\), so:\[RT = \sqrt{(5 - 1)^2 + (4 - 5)^2} = \sqrt{(4)^2 + (-1)^2} = \sqrt{16 + 1} = \sqrt{17}.\]

Compare Side Lengths

Compare the lengths of all sides, which are \(RS = 2\sqrt{5}\), \(ST = 3\sqrt{5}\), and \(RT = \sqrt{17}\). Notice that none of the side lengths are equal.

Conclude Triangle Type

Since none of the sides are of equal length, \(\triangle RST\) has three sides of different lengths, classifying it as a scalene triangle.

Key concepts

Distance Formula

The distance formula is a crucial tool in geometry, especially when working with the coordinates of points in a plane. It helps us determine the length between any two points when their coordinates are given. The formula comes from the Pythagorean theorem and can be expressed as:\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]This equation computes the straight-line distance, or "Euclidean distance," between two points \((x_1, y_1)\) and \((x_2, y_2)\) on a Cartesian plane. Let's break down this formula step by step:

• Subtract the x-coordinates \((x_2 - x_1)\) and y-coordinates \((y_2 - y_1)\) of the two points to find the differences.
• Square both the differences found above.
• Add these squared differences together.
• Finally, take the square root of this sum to find the distance.

The distance formula is especially helpful when working with triangles, as it allows for precise calculations of side lengths. This helps in classifying triangles by comparing the lengths of their sides.

Scalene Triangle

Triangles are categorized into several types, based on their side lengths. A scalene triangle is defined as a triangle with all three sides of different lengths. Therefore, it doesn't have any equal side lengths or equal angles. This uniqueness in side lengths gives the scalene triangle its distinctive look. To determine if a triangle is scalene, you must check the side lengths of the triangle.In our example, by applying the distance formula, we can calculate each side of the triangle:

• Side \( RS = 2\sqrt{5} \)
• Side \( ST = 3\sqrt{5} \)
• Side \( RT = \sqrt{17} \)

Since none of these side lengths are equal, the triangle \( \triangle RST \) is confirmed to be scalene. This method of side comparison is a reliable strategy for triangle classification, providing a clear understanding of the triangle's properties.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves using a coordinate plane to visualize and solve problems, making abstract concepts much more concrete. This branch of geometry blends algebra and geometry to enable the study of geometric figures using a numerical system.For triangle classification, coordinate geometry is indispensable because:

• It allows us to apply the distance formula to calculate side lengths accurately.
• It provides a visual approach, plotting points helps in understanding shapes and their properties better.
• Complex geometric problems can be simplified by converting them into algebraic equations.

In our task, the vertices \( R(1,5) \), \( S(-1,1) \), and \( T(5,4) \) are plotted on the coordinate plane. With the distance formula, we measure each side of the triangle, assisting in accurately determining its classification as scalene. Understanding coordinate geometry helps build a strong foundation, allowing you to tackle various geometric problems with confidence.