Problem 9 - Prealgebra

Question

Find the distance between each pair of points. Round to the nearest tenth, if necessary. $$S(-9,0), T(6,-7)$$

Step-by-Step Solution

Answer

The distance between S and T is approximately 16.6 units.

Step 1: Identify the Coordinates

First, identify the coordinates of points S and T. We have point S with coordinates \((-9, 0)\) and point T with coordinates \((6, -7)\).

Step 2: Apply the Distance Formula

To find the distance between two points, (x₁, y₁) and (x₂, y₂), use the distance formula: \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\].

Step 3: Substitute the Values

Substitute the coordinates of the points S and T into the distance formula: \[d = \sqrt{(6 - (-9))^2 + (-7 - 0)^2}\]. This simplifies to \[d = \sqrt{(6 + 9)^2 + (-7)^2}\].

Step 4: Calculate Inside the Square Root

Simplify inside the square root: \(d = \sqrt{(15)^2 + (-7)^2}\). This becomes \(d = \sqrt{225 + 49}\).

Step 5: Compute the Square Root

Calculate \( \sqrt{274} \). Use a calculator to find \(d ≈ 16.55294536\).

Step 6: Round the Value

Finally, round the calculated distance to the nearest tenth: \(d ≈ 16.6\).

Key Concepts

Coordinates

When working with points in a plane, we refer to them by their coordinates. This simply defines their exact location in a two-dimensional space, using two numbers placed in a pair. Each point is typically described as \(x, y\), where \(x\) represents the horizontal position and \(y\) represents the vertical position.
For the points in our exercise, S has coordinates \((-9, 0)\) and T has coordinates \(6, -7)\).

• \((-9, 0)\) tells us point S is 9 units left of the origin and on the x-axis.
• \(6, -7)\) places point T 6 units right and 7 units below the origin.

Understanding coordinates is crucial because they are the starting point for calculating distances and understanding geometric relationships.

Distance Between Points

The distance formula helps find the straight-line distance between two points on a coordinate plane. It's like measuring the length of a stick directly connecting the two points. This is based on the Pythagorean Theorem that forms a right triangle, allowing us to calculate the hypotenuse.
In our case, the formula is: \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]
For points S and T, we substituted the coordinates into the formula: S \((-9, 0)\) and T \(6, -7)\).

• We calculated the differences: \((6 - (-9)) = 15\) and \((-7 - 0) = -7\).
• Squaring both gives \(15^2 = 225\) and \((-7)^2 = 49\).
• Adding these, \((225 + 49 = 274)\), gives us the value to find the square root of.

This formula not only finds distances in geometry but also practically in fields like navigation and computer graphics.

Rounding Numbers

Rounding numbers is a handy skill for simplifying results to a more manageable form. In mathematics, especially when dealing with decimal numbers or measurements, precision can sometimes be adjusted for clarity or practicality.
In this exercise, the exact distance between points S and T was approximately 16.55294536. Rounding to the nearest tenth, look at the hundredths place. If it's 5 or more, you round up; if less, you round down.

• Here, after looking at the hundredths digit (5), we round up.
• The rounded distance becomes \(16.6\).

Employing rounding ensures numbers are easier to work with in everyday contexts or when reporting results in scientific, engineering, or mathematical fields.

Geometry

Geometry is the branch of mathematics that deals with shapes, sizes, and the properties of space. Understanding geometry allows us to manage different tasks, from simple measurements to complex calculations involving forms and spaces.
In the context of this exercise, knowing how to navigate a coordinate system and apply geometric formulas is essential. For example, determining the distance between two points on a plane is a basic but powerful geometric operation.

• It informs the concept of measurement in spaces that we see in construction and design.
• It's foundational in numerous applications, from mapping to designing digital graphics.

Grasping the principles of geometry helps understand not only spatial relationships but also the mathematical concepts underlying many aspects of the real world.