Problem 2 Find the distance between the po... [FREE SOLUTION] (2024)

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Chapter 11: Problem 2

Find the distance between the points. Write the answer in exact form and thenfind the decimal approximation, rounded to the nearest tenth if needed. $$ (-4,-3) \text { and }(2,5) $$

Short Answer

Expert verified

The distance is 10 in exact form and 10.0 as the decimal approximation.

Step by step solution

Identify the Points

Given points are $(-4, -3)$ and $(2, 5)$.

Apply the Distance Formula

Use the distance formula $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. In this case, $(x_1, y_1) = (-4, -3)$ and $(x_2, y_2) = (2, 5)$.

Compute the Differences

Calculate the differences: $x_2 - x_1 = 2 - (-4) = 6$ and $y_2 - y_1 = 5 - (-3) = 8$.

Square the Differences

Square the differences: $(x_2 - x_1)^2 = 6^2 = 36$ and $(y_2 - y_1)^2 = 8^2 = 64$.

Compute the Sum of Squares

Add the squares: $36 + 64 = 100$.

Find the Square Root

Take the square root of the sum: $d = \sqrt{100} = 10$.

Write the Answer in Exact Form

The exact distance is $10$.

Calculate Decimal Approximation

The decimal approximation for the distance is also $10.0$.

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

coordinate geometry

In coordinate geometry, we use a coordinate system to determine and describe the positions of points. Each point is defined by an ordered pair of numbers, usually written as $ (x, y) $, where $ x $ represents the point's horizontal position, and $ y $ represents its vertical position.
In the given problem, the points $ (-4, -3) $ and $ (2, 5) $ are provided. These coordinates help us understand where these points lie on the Cartesian plane, making it easier to visualize and calculate the distance between them using the distance formula.
Coordinate geometry is essential as it forms the basis for more advanced topics like graphing functions, transformations, and even calculus.

distance calculation

To find the distance between two points in a coordinate system, we use the distance formula. This formula is derived from the Pythagorean theorem. Let's break it down:
The distance formula is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
This formula calculates the straight-line distance between the two points $ (x_1, y_1) $ and $ (x_2, y_2) $ on the Cartesian plane.
Here's a step-by-step breakdown of how we applied this formula to our points $ (-4,-3) $ and $ (2, 5) $:

square root

The square root of a number is a value that, when multiplied by itself, gives the original number. It's a fundamental concept in mathematics, especially when dealing with distances and areas.
In our problem, after summing the squares of differences, we obtained 100. To find the actual distance, we took the square root of this sum:
\[ \sqrt{100} = 10 \]
This tells us that 10 is the number that, when squared, equals 100. Taking square roots helps convert the squared differences back into the same units as the original coordinates.
The notation for square root is often represented as \sqrt{}\. Practicing square roots helps in understanding not only geometry problems but various algebraic processes as well.

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Most popular questions from this chapter

Solve the system of equations by using substitution. $$ \left\\{\begin{array}{l} 9 x^{2}+y^{2}=9 \\ y=3 x+3 \end{array}\right. $$Graph each equation. $$ \frac{(y-3)^{2}}{9}-\frac{(x+2)^{2}}{16}=1 $$Graph each equation. $$ \frac{x^{2}}{9}+\frac{y^{2}}{25}=1 $$Graph each ellipse. $$ \frac{(x+1)^{2}}{4}+\frac{(y+6)^{2}}{25}=1 $$Graph. $$ 16 y^{2}-25 x^{2}=400 $$

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