Question

Find the distance \(d\) between the points \(P_{1}\) and \(P_{2}\). $$ P_{1}=(a, a) ; \quad P_{2}=(0,0) $$

Short answer

The distance is \( a \sqrt{2} \).

Step-by-step solution

Recall the distance formula

The distance formula between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) in a Cartesian plane is given by: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Substitute given points into the formula

Substitute \( (a, a) \) for \( P_1 \) and \( (0,0) \) for \( P_2 \) into the distance formula: \[ d = \sqrt{(0 - a)^2 + (0 - a)^2} \]

Simplify the expression inside the square root

Simplify the expression: \[ d = \sqrt{a^2 + a^2} \]

Combine like terms

Combine the terms under the square root: \[ d = \sqrt{2a^2} \]

Simplify the square root

Simplify the square root: \[ d = \sqrt{2} \cdot \sqrt{a^2} = \sqrt{2} \cdot a \]

Write the final answer

The distance \( d \) between the points \( (a, a) \) and \( (0,0) \) is: \[ d = a \sqrt{2} \]

Key concepts

Cartesian Plane

A Cartesian plane, also known as the coordinate plane, is a two-dimensional plane defined by a horizontal axis (the x-axis) and a vertical axis (the y-axis). These axes intersect at a point called the origin, which has coordinates (0,0). Any point on the Cartesian plane is represented by an ordered pair of numbers (x, y), where 'x' denotes the horizontal position, and 'y' denotes the vertical position.

Understanding the Cartesian plane is crucial for finding distances between points. Each point on the plane can move left or right along the x-axis and up or down along the y-axis. This grid system helps us visualize and calculate distances precisely.

Distance Between Points

To find the distance between two points on a Cartesian plane, we use the distance formula. The distance formula is derived from the Pythagorean theorem and is given by: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

In the given exercise, we need to find the distance between points \( P_1 = (a, a) \) and \( P_2 = (0,0) \). By substituting these points into the formula, we look at the change in x-coordinates, \( x_2 - x_1 \), and y-coordinates, \( (y_2 - y_1) \). Then, we square each difference, sum them, and finally, take the square root to get the distance.

This method helps to accurately determine how far apart two locations are on the plane.

Square Root Simplification

Simplifying square roots is a key step in solving distance problems. When you have an expression under a square root, like \( \sqrt{2a^2} \), you can often simplify it by breaking it into simpler parts.

In our example, we look at \( \sqrt{2a^2} \). We can separate this into \( \sqrt{2} \) and \( \sqrt{a^2} \). Since the square root of \( a^2 \) is just \( a \), the expression simplifies to: \[ \sqrt{2} \cdot a \]

This simplification helps us understand the distance in terms of simpler, easier-to-handle components, making it clearer and more intuitive.

Algebraic Expressions

Algebraic expressions include numbers, variables, and arithmetic operations. In distance calculations, we often work with formulas involving algebraic expressions.

For instance, in the problem, we start with the points \( (a, a) \) and \( (0, 0) \). Substituting these into the distance formula, we have: \[ d = \sqrt{(0 - a)^2 + (0 - a)^2} \] This involves performing operations inside the square root, simplifying it to combine like terms, and then further simplifying the square root itself.

Understanding how to manipulate algebraic expressions is crucial for breaking down complex problems into manageable steps, ultimately finding the simplest and most accurate answer. By practicing these operations, students develop stronger skills in handling various math problems.