Question

Find the distance between each pair of points. Round to the nearest tenth, if necessary. \(S(-6,-4), T(-3,-7)\)

Short answer

The distance between the points is approximately 4.2 units.

Step-by-step solution

Identify the Points

We are given the coordinates of two points: \(S(-6, -4)\) and \(T(-3, -7)\). These points will be used in our distance formula.

Recall the Distance Formula

The distance formula for points \((x_1, y_1)\) and \((x_2, y_2)\) is: \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\] We will use this formula to find the distance between the points \(S\) and \(T\).

Substitute the Coordinates into the Formula

Plug the coordinates of the points into the distance formula: \[d = \sqrt{((-3) - (-6))^2 + ((-7) - (-4))^2}\]This simplifies to: \[d = \sqrt{(-3 + 6)^2 + (-7 + 4)^2}\]

Simplify the Arithmetic

Simplify inside the parentheses: \[d = \sqrt{(3)^2 + (-3)^2}\]Now, calculate the squares: \[d = \sqrt{9 + 9}\]

Calculate the Square Root

Add the squared terms and then take the square root: \[d = \sqrt{18}\]Simplifying further: \[d \approx 4.2\] (rounded to the nearest tenth).

Key concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, blends algebra and geometry. This branch of mathematics uses the coordinate plane to describe shapes, sizes, and other properties of figures. In coordinate geometry, points are represented using ordered pairs \(x, y\), for example, \(S(-6, -4)\). Each point has a unique position on the plane that is determined by its x-coordinate and y-coordinate.

One of the essential uses of coordinate geometry is to calculate distances and slopes, and understand relationships between points and lines. With coordinate geometry, we can not only determine the properties of geometric figures but also solve problems algebraically, which would be difficult using geometry alone. This makes it a powerful tool in both mathematics and real-world applications.

Distance Calculation

In coordinate geometry, we can calculate the distance between two points on the Cartesian plane using the distance formula. This formula is derived from the Pythagorean theorem and can be written as:

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

For the given points \(S(-6, -4)\) and \(T(-3, -7)\), we substitute them into the formula:

• Subtract the x-coordinates: \(x_2 - x_1 = -3 + 6\)
• Subtract the y-coordinates: \(y_2 - y_1 = -7 + 4\)

After these substitutions, simplify the arithmetic inside the formula, first by calculating the squares: \[ ((3)^2 + (-3)^2 = 9 + 9) \]

Finally, calculate the square root of the sum: \[ d = \sqrt{18} \approx 4.2 \]

This result is then rounded to the nearest tenth, hence the distance between the points \(S\) and \(T\) is approximately 4.2 units. This approach can be applied to any pair of points on the Cartesian plane.

Cartesian Plane

The Cartesian plane, also known as the coordinate plane, is a two-dimensional surface defined by two perpendicular number lines: the x-axis (horizontal) and y-axis (vertical). These axes intersect at a point called the origin, which has coordinates (0, 0). The plane is divided into four quadrants, each quadrant representing different sign combinations of coordinates.

Points on the Cartesian plane are expressed using ordered pairs \(x, y\). The x-coordinate indicates how far the point is along the horizontal axis, while the y-coordinate shows the point's position along the vertical axis. This system allows for precise measurement and calculation of distances and angles between points. For example, to find the distance between \(S(-6, -4)\) and \(T(-3, -7)\), you use these coordinates in the distance formula, as described above.

The Cartesian plane is fundamental to coordinate geometry and serves as a basis for numerous mathematical concepts, such as graphing functions, calculating slopes, and exploring geometric transformations. It is a powerful tool that provides a visual and numerical way to analyze and interpret mathematical problems.