Problem 1 - Prealgebra

Question

Find the distance between each pair of points. Round to the nearest tenth, if necessary. A(-1,3), B(8,-6)

Step-by-Step Solution

Answer

The distance between A(-1,3) and B(8,-6) is approximately 12.7 units.

Step 1: Identify the formula

We need to use the distance formula to find the distance between two points in a plane. The distance formula is derived from the Pythagorean theorem and is given by: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points.

Step 2: Plug in the coordinates

Substitute the coordinates of points A and B into the distance formula. For point A(-1, 3) and point B(8, -6), we have:\[ x_1 = -1, \, y_1 = 3, \, x_2 = 8, \, y_2 = -6 \]Plug these into the formula:\[ d = \sqrt{(8 - (-1))^2 + (-6 - 3)^2} \]

Step 3: Simplify the expression

Calculate the differences in coordinates and then square them:\[ d = \sqrt{(8 + 1)^2 + (-6 - 3)^2} \]\[ d = \sqrt{9^2 + (-9)^2} \]\[ d = \sqrt{81 + 81} \]

Step 4: Calculate the distance

Add the squared differences:\[ d = \sqrt{162} \]Find the square root and round to the nearest tenth:\[ d \approx 12.7 \]

Step 5: Conclusion

The distance between points A(-1,3) and B(8,-6) is approximately 12.7 units when rounded to the nearest tenth.

Key Concepts

coordinates

In mathematics, coordinates are values that identify the precise location of a point on a plane or in space. When plotting a point in a 2-dimensional plane, coordinates are typically given as a pair of numbers:

• The first number is the x-coordinate, which indicates the horizontal position.
• The second number is the y-coordinate, which represents the vertical position.

For example, in the coordinates A(-1, 3), -1 is the x-coordinate and indicates the position left or right of the origin, which is (0,0), and 3 is the y-coordinate, which indicates the position up or down from the origin.
Coordinates are crucial in geometry as they allow us to determine positions and calculate distances between points. By understanding how to read and work with coordinates, you can utilize the distance formula effectively to find the length between two points.

Pythagorean theorem

The Pythagorean theorem is a fundamental principle in geometry, which explains the relationship between the sides of a right triangle. This theorem is given by the formula: \[ a^2 + b^2 = c^2 \]where:

• \(a\) and \(b\) are the lengths of the two shorter sides (legs) of the triangle.
• \(c\) is the length of the longest side (hypotenuse), opposite the right angle.

The distance formula, which calculates the distance between two points on a plane, is derived from the Pythagorean theorem. By treating the segment between two points like the hypotenuse of a right triangle, the theorem helps to determine the distance that connects them.
To find this distance, we treat the differences in their x-coordinates and y-coordinates as the triangle’s legs. Squaring these differences and summing them up matches the steps in the theorem, helping us get the result through square root calculation.

square root calculation

When calculating the distance using the distance formula, one of the essential steps is finding the square root of the sum of squared differences of the coordinates. The square root operation helps reverse the process of squaring, allowing us to solve for the actual distance or length of a line segment.
If we have a squared value such as \[ (8 + 1)^2 + (-6 - 3)^2 = 81 + 81 \]finding its square root gives you the length of the diagonal line connecting the two points, making it \[ \sqrt{162} \],which is calculated and rounded as approximately 12.7.

• Square root calculations are integral to transforming squared sums into distances that are more intuitive to understand and utilize practically.
• When working with decimals, rounding to the nearest tenth is common, ensuring the value is easy to work with in real-world applications.

Understanding how to calculate and interpret square roots is crucial in geometry and many practical problem-solving scenarios.