Problem 105 - A Graphical Approach to Precalculus with Limits

Question

Triangles can be classified by their sides. (a) An isosceles triangle has at least two sides of equal length. Determine whether the triangle with vertices \((0,0),(3,4),\) and \((7,1)\) is isosceles. (b) An equilateral triangle has all sides of equal length. Determine whether the triangle with vertices \((-1,-1),(2,3),\) and \((-4,3)\) is equilateral. (c) Determine whether a triangle having vertices \((-1,0),(1,0)\) and \((0, \sqrt{3})\) is isosceles, equilateral, or neither. (d) Determine whether a triangle having vertices \((-3,3),(-2,5)\) and \((-1,3)\) is isosceles, equilateral, or neither.

Step-by-Step Solution

Answer

(a) Isosceles, (b) Not equilateral, (c) Equilateral, (d) Isosceles.

Step 1: Calculate side lengths for (a)

To determine if the triangle with vertices \((0,0), (3,4), (7,1)\) is isosceles, we start by calculating the distances between each pair of points using the distance formula: \[d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\]Calculate the distances: 1. Between \((0,0)\) and \((3,4)\): \[d = \sqrt{(3-0)^2 + (4-0)^2} = \sqrt{9 + 16} = 5\]2. Between \((3,4)\) and \((7,1)\): \[d = \sqrt{(7-3)^2 + (1-4)^2} = \sqrt{16 + 9} = 5\]3. Between \((0,0)\) and \((7,1)\): \[d = \sqrt{(7-0)^2 + (1-0)^2} = \sqrt{49 + 1} = \sqrt{50}\]Since two sides are equal (both are 5), the triangle is isosceles.

Step 2: Calculate side lengths for (b)

Check whether the triangle with vertices \((-1,-1), (2,3), (-4,3)\) is equilateral. Compute the side lengths:1. Between \((-1,-1)\) and \((2,3)\): \[d = \sqrt{(2 + 1)^2 + (3 + 1)^2} = \sqrt{9 + 16} = 5\]2. Between \((2,3)\) and \((-4,3)\): \[d = \sqrt{(-4 - 2)^2 + (3 - 3)^2} = \sqrt{36} = 6\]3. Between \((-1,-1)\) and \((-4,3)\): \[d = \sqrt{(-4 + 1)^2 + (3 + 1)^2} = \sqrt{9 + 16} = 5\]Since not all sides are equal, the triangle is not equilateral.

Step 3: Classification of triangle (c)

Determine if the triangle with vertices \((-1,0), (1,0), (0,\sqrt{3})\) is isosceles, equilateral, or neither.Calculate the distances: 1. Between \((-1,0)\) and \((1,0)\): \[d = \sqrt{(1 + 1)^2 + (0 - 0)^2} = 2\]2. Between \((1,0)\) and \((0,\sqrt{3})\): \[d = \sqrt{(0 - 1)^2 + (\sqrt{3} - 0)^2} = \sqrt{4} = 2\]3. Between \((0,\sqrt{3})\) and \((-1,0)\): \[d = \sqrt{(-1 - 0)^2 + (0 - \sqrt{3})^2} = \sqrt{4} = 2\]All sides are equal, so the triangle is equilateral.

Step 4: Classification of triangle (d)

To classify the triangle with vertices \((-3,3), (-2,5), (-1,3)\), compute the side lengths:1. Between \((-3,3)\) and \((-2,5)\): \[d = \sqrt{(-2 + 3)^2 + (5 - 3)^2} = \sqrt{5}\]2. Between \((-2,5)\) and \((-1,3)\): \[d = \sqrt{(-1 + 2)^2 + (3 - 5)^2} = \sqrt{5}\]3. Between \((-3,3)\) and \((-1,3)\): \[d = \sqrt{(-1 + 3)^2 + (3 - 3)^2} = 2\]Since two of the sides are equal and one is different, the triangle is isosceles.

Key Concepts

Isosceles Triangle

An isosceles triangle is one of the basic categories in the classification of triangles based on their side lengths. To be classified as isosceles, a triangle must have at least two sides of equal length. This equality gives the triangle its name, derived from Greek words meaning 'equal legs'. In geometry, these equal-length sides are critical because they entail certain properties that are consistent and predictable, such as equal base angles opposite those sides.

When determining whether a triangle is isosceles, we can often utilize the distance formula to compare side lengths effectively. By checking for at least two sides with the same measure, we can conclusively state that a given triangle is isosceles. This geometric property is significant because it simplifies calculations and might reveal symmetry in the shape, which can be essential for solving various mathematical problems.

Equilateral Triangle

Equilateral triangles are special because all three of their sides are of equal length. This uniformity results in additional properties that make equilateral triangles quite fascinating. First and foremost, if all the sides are equal, then all the interior angles must also equal 60 degrees.

This makes equilateral triangles unique among triangles, as their symmetry is perfect. Not only are the sides equal, but the angles, as well as altitudes and medians, are congruent, creating a harmonious balance in shape and form.

• Each angle in an equilateral triangle is exactly 60 degrees.
• All internal properties, such as medians and altitudes, are equal in measure.

An equilateral triangle can be identified using the distance formula by checking whether each side of the triangle has equal length. If they do, this implies the triangle is equilateral, offering a useful visual and mathematical tool for problem-solving.

Distance Formula

The distance formula is a powerful tool used in geometry to determine the length between two points on a coordinate plane. It is derived from the Pythagorean theorem and is particularly useful in proving specific triangle properties. The formula is given by:

• \(d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\)

This formula calculates the hypotenuse of a right triangle formed by the horizontal (x) and vertical (y) distances, showing the actual straight-line distance between two points.

In problems like determining the type of triangle formed by connecting points on a plane, we use the distance formula to find the lengths of each side. We compare these calculated lengths to decide on the classification of the triangle, such as revealing if it is isosceles or equilateral. Using the distance formula helps simplify these complex geometric assessments.

Triangle Properties

Triangles, depending on their specific type, have a range of properties that can unlock insights into geometrical problems. Recognizing these properties can equip one with a better understanding of how to classify and utilize triangles effectively.

Some fundamental triangle properties include:

• The sum of the interior angles of any triangle is always 180 degrees.
• In an isosceles triangle, the angles opposite the equal sides are equal.
• In an equilateral triangle, all angles and sides are equal, with each angle measuring 60 degrees.

Understanding and applying these properties can vastly simplify solving geometric problems, such as computing unknown angles or verifying a triangle’s classification. Knowing these properties helps apply correct theorems and formulas efficiently.