91Ó°ÊÓ

In coordinate geometry, finding the distance between two points on a plane is a common exercise. This distance is symbolized as 'd' and is obtained using the distance formula. Imagine you have two points

• \( x_1, y_1 \ \)
• \( x_2, y_2 \ \)

The distance formula comes in handy to calculate 'd' between these points:\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \ \].So for example, if we need the distance 'd' between \( \sqrt{7}, 9 \sqrt{3} \ \) and \( -\sqrt{7}, 4\sqrt{3} \ \), substitution helps us evaluate the final answer. Calculations start by placing the values in the formula and simplifying step-by-step.

Understanding algebraic expressions is vital for solving distance-related problems. An algebraic expression is a combination of variables, numbers, and operations. For example, in the distance formula, expressions like \( \sqrt{7} \ \), \( 9\sqrt{3} \ \), and \( -\sqrt{7} \ \) come into play. **Simplifying Expressions** Simplify inputs before carrying out operations. For instance, from our exercise, we substitute directly: \( x_1 = \sqrt{7} \ \), \( y_1 = 9 \sqrt{3} \ \), \( x_2 = -\sqrt{7} \ \) and \( y_2 = 4 \sqrt{3} \ \). Simplify inside parentheses before further steps. For example, \( -\sqrt{7} - \sqrt{7} = -2 \sqrt{7} \ \) and \( 4 \sqrt{3} - 9 \sqrt{3} = -5 \sqrt{3} \ \).In algebra, consider associating and disassociating operations carefully.

Coordinate geometry is the study of geometric figures using a coordinate system. Here, we mainly focus on the Cartesian plane divided into four quadrants. Each point is defined by an ordered pair \( x, y \ \). When given two points, our task is to analyze their positional relation, often guided by the distance formula. **Using the Distance Formula in Coordinate Geometry**Our task is to apply the distance formula to points:

• \( (\sqrt{7}, 9 \sqrt{3}) \ \)
• \( ( -\sqrt{7}, 4 \sqrt{3} ) \ \)

The formula \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \ \) can be used to calculate geometric distances. Substitute the specific values, simplify, and then get the numeric distance: \( \sqrt{103} \ \).Coordinate geometry emphasizes such systematic methods to ensure accuracy.

Radicals are expressions containing a root symbol. Simplification involves reducing these radicals to their simplest form. The distance formula often results in such expressions, so understanding simplification principles is crucial. **Simplifying Step-by-Step**In the provided example, after substituting and simplifying, we get terms like: \( \sqrt{103} \ \). Simplification proceeds as:1. **Square the Terms** Sqaure each term inside the parentheses:

• Calculate: \( (-2 \sqrt{7} )^2\ \)
• Calculate: \( ( -5 \sqrt{3} )^2 \ \)

2. **Combine and Simplify**Finally, evaluating these will give: \sqrt{28 + 75} \ \}. Simplification aims to execute operations in logical steps leading to the simplest form, boosting understandability of further algebraic operations.