Problem 5 - Calculus Early Transcendentals: Pearson New International Edition

Question

Show that the triangle whose vertices are \((5,3),(-2,4)\), and \((10,8)\) is isosceles.

Step-by-Step Solution

Answer

The triangle is isosceles because two of its sides are equal.

Step 1: Find the Distance Between Each Pair of Vertices

To prove that the triangle is isosceles, we must find the lengths of all three sides of the triangle and identify if at least two are equal. Use the distance formula: \( d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \).

Step 2: Calculate the Distance Between Points \((5, 3)\) and \((-2, 4)\)

Apply the distance formula: \[ d = \sqrt{((-2) - 5)^2 + (4 - 3)^2} = \sqrt{(-7)^2 + (1)^2} = \sqrt{49 + 1} = \sqrt{50} = 5\sqrt{2}. \]

Step 3: Calculate the Distance Between Points \((-2, 4)\) and \((10, 8)\)

Apply the distance formula: \[ d = \sqrt{(10 - (-2))^2 + (8 - 4)^2} = \sqrt{12^2 + 4^2} = \sqrt{144 + 16} = \sqrt{160} = 4\sqrt{10}. \]

Step 4: Calculate the Distance Between Points \((5, 3)\) and \((10, 8)\)

Apply the distance formula: \[ d = \sqrt{(10 - 5)^2 + (8 - 3)^2} = \sqrt{5^2 + 5^2} = \sqrt{25 + 25} = \sqrt{50} = 5\sqrt{2}. \]

Step 5: Compare the Side Lengths

The side lengths are \(5\sqrt{2}\), \(4\sqrt{10}\), and \(5\sqrt{2}\). Since two sides are equal \(5\sqrt{2} = 5\sqrt{2}\), the triangle is isosceles.

Key Concepts

Distance Formula

The distance formula is a tool used for determining the distance between two points in a Cartesian plane. This is essential in coordinates geometry when you want to calculate the length of a line segment connecting two points. The points are defined by their coordinates, such as

• Point A: egin{equation}(x_1, y_1) onumberegin{equation}
• Point B: egin{equation}(x_2, y_2)egin{equation}

To find the distance between these points, the formula is:\[d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\]Each pair of coordinates is plugged into the formula, signifying the changes along the x-axis and y-axis. This formula is derived from the Pythagorean Theorem. When you calculate the square root of the sum of the squares of these changes, you find the direct distance across the plane between the points. This method is crucial in analyzing the geometry of triangles to determine side lengths, which are key to identifying triangle types, such as an isosceles triangle.

Coordinates Geometry

Coordinates geometry, also known as analytic geometry, merges algebra and geometry using coordinates and formulas.
This branch of mathematics allows for a more precise analysis of geometric figures through the use of a coordinate system and numerical calculations.
In our specific problem:

• The coordinates for the triangle’s vertices are given as egin{equation}(5,3) onumber egin{equation}, egin{equation}(-2,4) onumber egin{equation}, and egin{equation}(10,8) onumber egin{equation}.
• Each vertex is a point in the two-dimensional plane that helps define the shape and size of the triangle.
• By applying the distance formula here, we can determine the side lengths.

Finding and using these coordinates ensures accurate calculations and conclusions about geometric figures, such as verifying the type of triangle given specific point locations.

Triangle Properties

Understanding the properties of a triangle is important when analyzing a triangle's type and characteristics. An isosceles triangle, for instance, is defined by having at least two sides that are equal in length, leading to two angles inside the triangle also being equal.
Here's a quick breakdown of triangle properties:

• An isosceles triangle always has two equal sides.
• The angles opposite the equal sides are equal.
• The unequal side, if present, is called the 'base' of the isosceles triangle.

In examining the vertices given in the exercise and calculating the side lengths using the distance formula, we discovered two sides were equal, indicating an isosceles triangle. Identifying these properties helps us understand that not only are the sides significant, but the angles and symmetry of the triangle play a crucial role in its geometric identity. Being comfortable with these concepts enhances your abilities to confidently solve similar geometric problems.