Problem 45 - Precalculus with Limits

Question

In Exercises 43-46, show that the points form the vertices of the indicated polygon. Isosceles triangle: \( (1, -3) \), \( (3, 2) \), \( (-2, 4) \)

Step-by-Step Solution

Answer

Yes, the given points (\(1, -3\)), (\(3, 2\)), and (\(-2, 4\)) can indeed form the vertices of an isosceles triangle.

Step 1: Calculate the distance between points

The Euclidean distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) in a plane is given by the formula:\[d = \sqrt{{(x_2 - x_1)}^{2} + {(y_2 - y_1)}^{2}}\]Using this information, calculate the distances between the given points:\(d_{13} = \sqrt{{(1 - (-2))}^{2} + {(-3 - 4)}^{2}} = \sqrt{{3}^{2} + {-7}^{2}} = \sqrt{9 + 49} = \sqrt{58}\)\(d_{12} = \sqrt{{(1 - 3)}^{2} + {(-3 - 2)}^{2}} = \sqrt{{-2}^{2} + {-5}^{2}} = \sqrt{4 + 25} = \sqrt{29}\)\(d_{23} = \sqrt{{(3 - (-2))}^{2} + {(2 - 4)}^{2}} = \sqrt{{5}^{2} + {-2}^{2}} = \sqrt{25 + 4} = \sqrt{29}\)

Step 2: Determine if it's an isosceles triangle

An isosceles triangle has two sides of equal length. Comparing \(d_{13}\) , \(d_{12}\), and \(d_{23}\), it's visible that \(d_{12} = d_{23} = \sqrt{29}\) which are equal. Thus, the points indeed form the vertices of an isosceles triangle.

Key Concepts

Euclidean Distance

Euclidean distance is a term that many students encounter in geometry and is vital for understanding how to find the distance between two points in a plane. It's like the straight-line distance between two spots on a map. When dealing with coordinates, you can think of it as how a crow would fly from one point to another—the most direct route.

To figure out this distance, mathematicians have developed a formula that uses the coordinates of the two points. It's very much like following a path directly from one point to another, without any turns or detours. In the context of our isosceles triangle exercise, Euclidean distance helped us to determine the lengths of the sides of the triangle.

Distance Formula

Understanding the distance formula is a bit like learning a secret code to unlock the lengths of lines on a grid. This formula is based on the Pythagorean theorem, which you might recall is about right triangles. But we can use it for any two points, not just the corners of a right triangle.

Here's the distance formula again for reference: \[d = \sqrt{{(x_2 - x_1)}^{2} + {(y_2 - y_1)}^{2}}\]
Once we plug in the coordinates of two points into this formula, the result gives us the length of the line connecting them. In our exercise, by applying the formula, we could tell whether the sides of our triangle were equal or not, confirming if it was isosceles.

Coordinates of a Point

The coordinates of a point are like an address for a location on a grid, with the 'x' value telling us how far to move to the left or right and the 'y' value how far to go up or down. In the exercise, the coordinates were used as vital inputs for the distance formula.

Remember, the 'x' value is always the first number and tells us the point's horizontal position, while the 'y' value is second and gives the vertical position. Using the coordinates, we can pinpoint the exact location of each vertex of our triangle on the plane.

Properties of Isosceles Triangles

Isosceles triangles are fascinating shapes! These triangles are special because they always have at least two sides that are exactly the same length. It's like having a pair of twin sides. This unique property is what gives an isosceles triangle its name and sets it apart from other triangles.

Knowing the properties of an isosceles triangle was crucial for the solution; it allowed us to conclude that since two of the triangle’s sides were equal in length, as seen with the distances calculated, our triangle was indeed an isosceles. An interesting extra property is that the angles opposite those equal sides are also equal—which helps further in understanding the symmetry and balance in isosceles triangles.