Problem 29 - College Algebra with Modeling and Visualization

Question

An isosceles triangle has at least two sides of equal length. Determine whether the triangle with vertices \((0,0),(3,4),(7,1)\) is isosceles.

Step-by-Step Solution

Answer

The triangle is isosceles with two sides of equal length 5.

Step 1: Calculate the Distance Between Vertices (0,0) and (3,4)

To find the distance between two points \((x_1, y_1)\) and \((x_2, y_2)\), we use the distance formula \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\].For points \((0,0)\) and \((3,4)\), this becomes:\[d = \sqrt{(3 - 0)^2 + (4 - 0)^2} = \sqrt{9 + 16} = \sqrt{25} = 5.\]

Step 2: Calculate the Distance Between Vertices (3,4) and (7,1)

Using the distance formula with points \((3,4)\) and \((7,1)\):\[d = \sqrt{(7 - 3)^2 + (1 - 4)^2} = \sqrt{4^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5.\]

Step 3: Calculate the Distance Between Vertices (0,0) and (7,1)

Using the distance formula with points \((0,0)\) and \((7,1)\):\[d = \sqrt{(7 - 0)^2 + (1 - 0)^2} = \sqrt{49 + 1} = \sqrt{50} = 5\sqrt{2}.\]

Step 4: Compare the Distances

We have calculated the distances:- Between \((0,0)\) and \((3,4)\): 5- Between \((3,4)\) and \((7,1)\): 5- Between \((0,0)\) and \((7,1)\): \(5\sqrt{2}\)Since the first two distances are equal, the triangle is isosceles.

Key Concepts

Distance Formula

The distance formula is an essential tool in coordinate geometry. It helps calculate the straight-line distance between two points in a plane. This formula is derived from the Pythagorean theorem and is written as:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]Here,

• \(x_1, y_1\) and \(x_2, y_2\) are the coordinates of the two points.
• The differences \((x_2 - x_1)^2\) and \((y_2 - y_1)^2\) are squared, summed up and taken under the square root.

This formula gives us the length of the segment connecting the two points, represented as a straight line, regardless of its orientation in the plane. It's frequently used in problems involving shapes and configurations on the coordinate grid.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, combines algebra and geometry to solve geometry problems using a coordinate system. In this system, geometric figures like lines, points, and shapes are represented using coordinates, which are usually pairs \((x, y)\) in a two-dimensional space.
Using this method, it becomes much easier to analyze geometric shapes. Here is how:

• Each point is given specific coordinates, making it simpler to calculate distances or midpoints between them using formulas.
• Transformations like translations and reflections are easier to understand and perform.
• Geometric properties, such as collinearity of points or parallelism of lines, are tested using algebraic equations derived from the coordinates.

Coordinate geometry enables precise and clear planning when working with figures, providing a straightforward approach to analyzing and solving geometric problems.

Triangles Analysis

Analyzing triangles involves examining their sides' lengths, angles, and properties. In this case, we are checking if a triangle with given vertices is isosceles. An isosceles triangle has at least two equal sides.
Steps to determine isosceles property:

• Calculate the distance between each pair of vertices using the distance formula.
• Compare these lengths. If two are equal, the triangle is isosceles.

Using this approach, we find two sides have equal lengths:

• The side from \((0,0)\) to \((3,4)\)
• The side from \((3,4)\) to \((7,1)\)

Both measured 5 units, affirming the triangle's isosceles nature. Triangle analysis helps us understand and confirm the type of triangle we are working with, leading to accurate conclusions in problems.