Problem 13 - Algebra 1: Concepts and Skills

Question

Find the distance between the two points. Round your solution to the nearest hundredth if necessary. $$ (5,8),(-2,3) $$

Step-by-Step Solution

Answer

The distance between the two points (5,8) and (-2,3) is approximately 8.60 units.

Step 1: Identify the coordinates

We are given two points (5,8) and (-2,3). Here, (x1, y1) = (5, 8) and (x2, y2) = (-2, 3).

Step 2: Substitute thevalues into the distance formula

Now plug these coordinates into the distance formula \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\), which results in \(d = \sqrt{((-2 - 5)^2 + (3 - 8)^2)}\).

Step 3: Simplify the equation

The calculation simplifies to \(d = \sqrt{(7^2) + (-5^2)}\), which further simplifies to \(d = \sqrt{49 + 25} = \sqrt{74}\)

Step 4: Convert to Decimal

Convert the result into decimal form, rounding to the nearest hundredth as stated in the question. This results in \(d \approx 8.60\)

Key Concepts

Understanding the Coordinate Plane

The coordinate plane is an essential tool in mathematics for visually representing and analyzing geometric figures. It consists of two number lines: the horizontal axis (x-axis) and the vertical axis (y-axis) which intersect at a point called the origin, marked as (0,0). Every point on the plane can be represented as a pair of numbers known as coordinates, which describe its position in relation to the origin, with the first number (the x-coordinate) indicating the horizontal position, and the second number (the y-coordinate) indicating the vertical position.

For example, the point (5,8) lies 5 units to the right of the origin on the x-axis and 8 units above the origin on the y-axis. Conversely, the point (-2,3) lies 2 units to the left and 3 units above the origin. The coordinates allow us to plot precise locations and measure distances between points to solve problems such as the one given.

The Role of the Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry, and it's incredibly useful for calculating distances on the coordinate plane. It tells us that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. The theorem is expressed mathematically as \(c^2 = a^2 + b^2\), where \(c\) is the hypotenuse, while \(a\) and \(b\) represent the other two sides.

When dealing with points on the coordinate plane, the distance between two points forms the hypotenuse of a right triangle, with the horizontal and vertical differences between the points as the other two sides. By applying the Pythagorean theorem, we can derive the distance formula used to find the distance between the two points.

Square Roots and Their Significance

Square roots are mathematical operations that answer the question, 'What number times itself gives me this value?' The square root of a number \(x\) is denoted as \(\sqrt{x}\) and represents a value that, when squared, equals \(x\). For instance, since \(7^2 = 49\), the square root of 49 is 7, represented as \(\sqrt{49} = 7\).

In the context of finding distances, after applying the Pythagorean theorem, we take the square root of the sum of squares to find the length of the hypotenuse. In our problem, this means finding \(\sqrt{74}\), which is a crucial step in the calculation that ultimately leads to the distance between the two points.

Decimal Rounding Techniques

Decimal rounding is the process of adjusting a decimal number to a certain number of desired decimal places for simplicity, clarity, or to align with a given precision requirement. To round to the nearest hundredth, for example, we look at the third decimal place. If this digit is 5 or greater, we round up the second decimal place by one. If it's less than 5, we leave the second decimal place as is.

In the solution to the problem, the square root of 74 is an irrational number and it goes on forever. To present this in a useful and readable manner, we round it to the nearest hundredth, which gives us an approximate value of 8.60. Rounding helps in reporting measurements in practical real-world applications where absolute precision isn't necessary.