Question

Find the distance between each pair of points. Round to the nearest tenth, if necessary. \(D(-1,-2), E(-3,-4)\)

Short answer

The distance between points \(D\) and \(E\) is approximately 2.8 units.

Step-by-step solution

Identify the Points

To solve the problem, we first identify the coordinates of the two given points. The coordinates of point \(D\) are \((-1, -2)\) and those of point \(E\) are \((-3, -4)\).

Use the Distance Formula

The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the distance formula: \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).

Substitute the Coordinates into the Formula

Substitute the point coordinates into the distance formula. Here, \(x_1 = -1\), \(y_1 = -2\), \(x_2 = -3\), and \(y_2 = -4\). This gives: \[d = \sqrt{(-3 - (-1))^2 + (-4 - (-2))^2}\].

Simplify the Differences

Calculate the differences inside the parentheses: \((-3 - (-1)) = (-3 + 1) = -2\) and \((-4 - (-2)) = (-4 + 2) = -2\). Substitute these back into the formula: \[d = \sqrt{(-2)^2 + (-2)^2}\].

Calculate the Squares

Calculate the squares of the differences: \((-2)^2 = 4\). Substitute into the equation: \[d = \sqrt{4 + 4}\].

Compute the Square Root Sum

Add the results: \(4 + 4 = 8\). Then, find the square root, \(\sqrt{8}\), which simplifies to \(2\sqrt{2}\) or approximately \(2.8\).

Final Answer

Round the result to the nearest tenth, which gives us a final answer of \(2.8\).

Key concepts

distance formula

The distance formula is a mathematical tool used to find the distance between two points in a coordinate plane. - It is derived from the Pythagorean theorem, which is a fundamental principle for calculating distances. - The distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula: \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\] - This formula effectively creates a right triangle where the differences in the x-coordinates and y-coordinates represent the two legs. - The line connecting the two points is the hypotenuse. This is why the Pythagorean theorem relates directly to the distance formula. Using the distance formula is straightforward: - First, identify the coordinates of each point. - Substitute these values into the formula. - Simplify the expression by calculating the differences and their squares. - Finally, take the square root of the sum to determine the distance. Understanding this concept will help you solve various problems in geometry efficiently.

coordinate geometry

Coordinate geometry, also known as analytic geometry, is the study of geometric objects using a coordinate system. - Points are described using their coordinates, which are numerical values that represent a specific location on a plane. - The most common system uses two axes: the x-axis (horizontal) and the y-axis (vertical). This method allows us to compute various geometric properties: - Location and position of points. - Distances between points, lines, and shapes. - Slopes of lines and angles between them. In the exercise you are dealing with, each point has a specific set of coordinates, such as - Point D declared as \((-1, -2)\) - Point E declared as \((-3, -4)\) With these coordinates, you can easily perform mathematical operations like those required to use the distance formula. Coordinate geometry is the bridge between algebra and geometry. It uses algebraic techniques to resolve geometric problems and is widely used in various branches of mathematics and engineering. By growing comfortable with coordinate geometry, you will be well-prepared to tackle complex spatial problems.

rounding to nearest tenth

Rounding to the nearest tenth is a basic math skill that simplifies a number to only one decimal place. This is crucial for presenting results in a cleaner, more understandable format. Here's how you do it:- Look at the number in the hundredths place (the second decimal place).- If the number is 5 or more, you round up the number in the tenths place.- If the number is less than 5, you keep the number in the tenths place the same. Let's use our calculation from the exercise to illustrate this: The distance calculated was approximately \(2.828\) - Here, the number in the tenths place is 8, and the hundredths place is 2.- Since 2 is less than 5, the 8 remains unchanged.Thus, the final answer is rounded to \(2.8\). Rounding is important in various fields, from engineering to finance, where precision to a certain decimal is vital for practical purposes. Mastering rounding will make your mathematical work both precise and approachable.