Problem 65 - Algebra 2

Question

GEOMETRY Find the perimeter of a triangle with vertices at \((2,4),(-1,3)\) and \((1,-3) .(\text { lesson } 10-1)\)

Step-by-Step Solution

Answer

The perimeter is the sum of \( \sqrt{10} \), \( \sqrt{40} \), and \( \sqrt{50} \).

Step 1: Understand the Problem

We need to find the perimeter of a triangle with the given vertices. The perimeter is the sum of the lengths of all sides of the triangle. Each side's length can be determined using the distance formula between two points in the coordinate plane.

Step 2: Recall the Distance Formula

The distance between two points \(x_1, y_1\) and \(x_2, y_2\) is calculated using the formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] We will apply this formula to calculate each side of the triangle.

Step 3: Calculate Side AB

Vertices A and B are given as (2, 4) and (-1, 3), respectively. Using the distance formula:\[ d_{AB} = \sqrt{((-1) - 2)^2 + (3 - 4)^2} = \sqrt{(-3)^2 + (-1)^2} = \sqrt{9 + 1} = \sqrt{10} \]

Step 4: Calculate Side BC

Vertices B and C are (-1, 3) and (1, -3), respectively. Using the distance formula:\[ d_{BC} = \sqrt{(1 + 1)^2 + (-3 - 3)^2} = \sqrt{2^2 + (-6)^2} = \sqrt{4 + 36} = \sqrt{40} \]

Step 5: Calculate Side CA

Vertices C and A are (1, -3) and (2, 4), respectively. Using the distance formula:\[ d_{CA} = \sqrt{(2 - 1)^2 + (4 + 3)^2} = \sqrt{1^2 + 7^2} = \sqrt{1 + 49} = \sqrt{50} \]

Step 6: Add to Find the Perimeter

The perimeter of a triangle is the sum of the lengths of its sides. \[ \text{Perimeter} = d_{AB} + d_{BC} + d_{CA} = \sqrt{10} + \sqrt{40} + \sqrt{50} \]

Key Concepts

Distance Formula

The distance formula is a crucial tool in coordinate geometry, especially when dealing with problems involving the measurement of distances between points. It helps to determine how far apart two points are from each other in a plane. The formula is given by

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

This expression involves squaring the differences of the corresponding coordinates of the two points and then taking the square root of their sum.
By applying this formula, you can calculate the precise length of a side of a triangle when its vertices are known.
Understanding the formula involves knowing that the square of differences corresponds to Pythagorean principles, reflecting the distance in the Cartesian coordinate system.

Coordinate Geometry

Coordinate geometry, also known as analytical geometry, is the field of geometry where we study geometry using a coordinate system. This discipline plays a centralized role in modern geometry, serving as its bridge to algebra.
By utilizing coordinates, you can represent geometric figures in the plane using algebraic equations.
For example, a triangle in coordinate geometry is described by the points where its vertices lie. Each vertex has a unique pair of coordinates \((x, y)\).
The main advantage of coordinate geometry is its ability to provide precise methods for calculating properties such as distances and slopes. When the triangle vertices are known, it becomes straightforward to apply formulas for perimeter, area, and other properties.

Triangle Vertices

In any triangle, the vertices represent the three corner points that define its shape and orientation in a plane.
When dealing with a triangle in coordinate geometry, these vertices are expressed as coordinates, typically denoted as points such as A, B, and C with values like \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\).

• Vertices provide the fundamental data needed for all sorts of geometric calculations, including determining the side lengths.
• Knowing the coordinates allows for the application of various formulas, such as those for distance or area, without needing additional information.

When you have the vertices of a triangle, you can easily use mathematical tools to calculate key features like its perimeter or area.

Perimeter Calculation

Finding the perimeter of a triangle involves adding the lengths of its three sides. In the context of coordinate geometry, these lengths are calculated using the distance formula for each side between two vertices.
Here are the steps to find the perimeter:

• Use the distance formula to calculate the length of the first side by plugging in the coordinates of the first and second vertices.
• Repeat this for the second side with the second and third vertices.
• Lastly, calculate the length of the third side using the coordinates of the third and first vertices.

Once all side lengths are determined, simply sum them up. This provides the total perimeter of the triangle, giving you insight into the triangle’s size and boundaries in a coordinate plane.