Problem 56 - Calculus: One and Several Variables

Question

The points are the vertices of a triangle. State whether the triangle is isosceles (two sides of equal length). a right triangle, both of these, or neither of these. $$P_{0}(-1,2) . \quad P_{1}(1,3) . \quad P_{2}(4,1)$$

Step-by-Step Solution

Answer

To determine whether the triangle formed by points \(P_{0}(-1,2)\), \(P_{1}(1,3)\), and \(P_{2}(4,1)\) is isosceles, a right triangle, both, or neither, perform the following calculations and comparisons: calculate the distances between the points using the distance formula, compare the distances to determine if the triangle is isosceles, and square the distances and compare to check if the triangle is a right triangle.

Step 1: Calculate pairwise distances

Using the distance formula \(d = \sqrt{(x_{2}-x_{1})^{2} + (y_{2}-y_{1})^{2}}\), calculate the distances between \(P_{0}\) and \(P_{1}\), \(P_{0}\) and \(P_{2}\), and \(P_{1}\) and \(P_{2}\).

Step 2: Determine if the triangle is isosceles

Compare the three distances. If any two are equal, the triangle is isosceles.

Step 3: Determine if the triangle is a right triangle

Square all the distances obtained in Step 1. If the largest squared distance is equal to the sum of the squares of the other two distances, the triangle is a right triangle. Noticed that the triangle can be both isosceles and right angle at the same time in which it is called an isosceles right triangle.

Key Concepts

Distance Formula

Understanding the distance formula is crucial when studying the geometry of triangles. It provides a method to calculate the exact length between two points in a coordinate system, which is essential for classifying the types of triangles.

The distance formula is derived from the Pythagorean theorem and in two dimensions, it is expressed as: \[d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\]Where \((x_1, y_1)\) and \((x_2, y_2)\) represent the coordinates of the two points. This formula helps determine the side lengths of a triangle when the vertices are given, enabling us to further classify the triangle as isosceles, equilateral, or scalene.

To ensure the concepts are easier to grasp, it's helpful to visualize the formula in application. Imagine you have two points on a grid and you want to find out how far apart they are in a straight line; this is similar to imagining a line segment connecting these points in space. By substituting the coordinates of these points into the formula, you calculate the length of this segment.

Isosceles Triangle

In triangle geometry, an isosceles triangle holds a special place. It is characterized by having at least two sides of equal length. These equal sides are known as the 'legs' of the triangle, and the unequal side is referred to as the 'base'.

An isosceles triangle also has two angles that are the same, which are the angles opposite the equal sides. This makes it simple to recognize an isosceles triangle by either its side lengths or its angle measurements.

When working with coordinates and the distance formula, you can verify if a triangle is isosceles by calculating the lengths of its sides. If any two sides have equal length, then the triangle fits the criteria of being isosceles. This aspect is fundamental when assessing triangles as it brings us a step closer to understanding their properties and potential symmetry.

Right Triangle

A right triangle, another key figure in triangle geometry, has one angle that measures precisely 90 degrees. This is often called the 'right angle'. Identifying a right triangle involves the use of the Pythagorean theorem which states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

In a coordinate system, if we've labeled the lengths of the sides as a, b, and c (with c being the hypotenuse), we must find that \(c^2 = a^2 + b^2\) for a triangle to be classified as a right triangle. If working with vertices and using the distance formula, comparing the squared distances can reveal whether a triangle is right-angled without the need to measure any angles directly.

Understanding this is significant because right triangles are the foundation of trigonometry and play a crucial role in various scientific and engineering applications. They are also a fundamental component for proving further properties in more complex geometric shapes.

Triangle Geometry

Triangle geometry encompasses the study of shapes with three sides and three angles. Classifying triangles relies on understanding their sides and angles. There are several types of triangles, including isosceles, equilateral, scalene, and right-angled triangles.

Each type of triangle has unique properties that define its angles and sides. An equilateral triangle has all three sides of equal length. A scalene triangle has no sides of equal length. Recognizing the various properties of triangles enhances our ability to solve geometric problems and understand the world around us.

When evaluating triangle problems, remember that a triangle's internal angles always add up to 180 degrees, which is vital when using angle measurements to classify a triangle. By exploring different triangles through the lens of their geometric properties, students can deepen their comprehension of more complex mathematical concepts and their applications in real-world situations.