Problem 55 - 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. $$P_{0}(-2,5), \quad P_{1}(1,3) . \quad P_{2}(-1,0)$$

Step-by-Step Solution

Answer

The triangle is isosceles but not right-angled.

Step 1: Calculate the Lengths of the Sides

Let's name the sides using two points: \[AB = P_{0}P_{1} = sqrt{((1 - (-2))^2 + (3 - 5)^2)} = sqrt{9 + 4} = sqrt{13}\] \[BC = P_{1}P_{2} = sqrt{((-1 - 1)^2 + (0 - 3)^2)} = sqrt{4 + 9} = sqrt{13}\] \[AC = P_{0}P_{2} = sqrt{((-1 - (-2))^2 + (0 - 5)^2)} = sqrt{1 + 25} = sqrt{26}\]

Step 2: Evaluate the Triangle Type Based on the Lengths of the Sides

Two sides (AB & BC) of the triangle have equal lengths \(sqrt{13}\). Hence the triangle is isosceles. Let's check if it's right-angled by applying the Pythagorean theorem. The longest side is AC, so we check if \(AC^2 = AB^2 + BC^2\), or \(sqrt{26}^2 = sqrt{13}^2 + sqrt{13}^2\). Substituting, we get 26 ≠ 2*13, so the triangle is not a right triangle.

Key Concepts

Isosceles Triangle

An isosceles triangle is a fascinating shape in geometry, having two sides of equal length. This unique feature also affects its angles, as the angles at the base of the equal sides are always of the same measure, which is an important clue when examining the properties of a triangle. In the exercise, we are given the coordinates of the vertices of a triangle.

By computing the distance between the points, we find that two sides are of equal length, precisely \( \sqrt{13} \). This clearly categorizes the triangle as isosceles, as it satisfies the fundamental condition of having at least two equal sides.

Isosceles triangles are widely studied because of their symmetry and appear frequently in various fields including architecture and engineering, making understanding them incredibly beneficial.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in Euclidean geometry that relates the lengths of the sides in a right triangle. According to this theorem, in any right triangle, 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. The theorem is typically expressed as \( c^2 = a^2 + b^2 \).

In our problem, we try to apply the theorem to determine if the triangle with sides \( \sqrt{13}, \sqrt{13} \), and \( \sqrt{26} \) is right-angled. By attempting to apply the theorem \( \sqrt{26}^2 = \sqrt{13}^2 + \sqrt{13}^2 \) and assessing the resultant equation, we determine that the condition does not hold, hence the triangle is not right-angled. This illustrates the theorem's utility in classifying triangles and its omnipresence in mathematical problems dealing with shapes and space.

Distance Formula

The distance formula is an application of the Pythagorean theorem in coordinate geometry, used to calculate the distance between two points in a plane. The general form of the distance formula is derived from the coordinates of the points \( (x_1, y_1) \) and \( (x_2, y_2) \) as follows: \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \).

This formula is crucial in our exercise, as it assists us in computing the lengths of the sides of the triangle by taking the vertices' coordinates as input. For instance, to determine the length between the points \( P_0(-2,5) \) and \( P_1(1,3) \) we've used the formula which resulted in \( \sqrt{13} \).

The distance formula thus bridges the gap between algebra and geometry, enabling us to analyze spatial relationships using algebraic methods.

Right Triangle

A right triangle is another basic geometric shape characterized by one angle measuring 90 degrees, known as the right angle. The side opposite this angle is the hypotenuse and it's the longest side of the right triangle. Right triangles play a significant role in various fields, from trigonometry to real-world applications such as construction and navigation.

To identify a right triangle, the Pythagorean theorem is used as a verification tool. If a triangle's side lengths satisfy this condition, it's categorized as a right triangle. As we've seen in the provided exercise, even though the triangle had two equal sides, it did not fulfill the criteria to be a right triangle; the square of the longest side did not equal the sum of the squares of the other two sides. Understanding the properties of right triangles is critical for problem-solving and analytical thinking in geometric contexts.