Problem 24 - Algebra 1: Concepts and Skills

Question

Find the distance between the two points. Round your solution to the nearest hundredth if necessary. $$ (-10,-2),(1,7) $$

Step-by-Step Solution

Answer

The distance between the points (-10,-2) and (1,7) is approximately 14.21 units.

Step 1: Identify the given points

The two points given are (-10,-2) and (1,7). We'll denote them as (x1, y1) and (x2, y2) respectively. So, x1 = -10, y1 = -2, x2 = 1, and y2 = 7.

Step 2: Substitute the values into the distance formula

Substitute the coordinates of the given points into the distance formula, which is \(\sqrt{{(x_2-x_1)^2 + (y_2-y_1)^2}}\). We get: \(\sqrt{{(1 - (-10))^2 + (7 - (-2))^2}}\).

Step 3: Perform the operations

Perform the calculations inside the square root expression. We get: \(\sqrt{{(1 + 10)^2 + (7 + 2)^2}}\), which simplifies to \(\sqrt{{11^2 + 9^2}}\), and then to \(\sqrt{{121 + 81}}\), and finally to \(\sqrt{202}\).

Step 4: Simplify the square root

The square root of 202 is approximately 14.21 when rounded to the nearest hundredth.

Key Concepts

Coordinate Plane

The coordinate plane is a two-dimensional surface on which we can plot points, lines, and curves using an x (horizontal) and y (vertical) axis. Each point on the plane is defined by an ordered pair of numbers, known as coordinates, which are typically written in the form \( (x, y) \). For instance, the two points in our exercise, \( (-10, -2) \) and \( (1, 7) \) are located by moving horizontally to -10 and vertically to -2 for the first point, and horizontally to 1 and vertically to 7 for the second point, respectively.

Understanding the coordinate plane is essential in various fields such as mathematics, physics, engineering, and computer science, as it allows for precise graphical representation of mathematical concepts and relationships.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry that relates the lengths of the sides of a right triangle. It states that in a right 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 other two sides. The formula is expressed as \( c^2 = a^2 + b^2 \), where \( c \) is the hypotenuse, and \( a \) and \( b \) are the other two sides.

In the context of the coordinate plane, we often use the theorem to find the distance between two points by treating the difference in the x-coordinates and y-coordinates as the sides of a right triangle, and the distance between the points as the hypotenuse.

Square Root Operations

Square root operations are mathematical functions that find a number which, when multiplied by itself, will yield the original number. For example, the square root of 9 is 3, because \( 3^2 = 9 \). The square root is symbolized by \( \sqrt{} \), and finding the square root is crucial when we want to solve equations involving squares or when applying the Pythagorean theorem in the distance formula. It essentially 'undoes' the squaring process.

In our exercise, after employing the Pythagorean theorem to express the distance as the square of the hypotenuse, we perform a square root operation to discover the actual length of the distance.

Calculating Distance

Calculating the distance between two points on the coordinate plane involves combining the concepts we've discussed. We start with the distance formula, which is derived from the Pythagorean theorem: \( \sqrt{{(x_2-x_1)^2 + (y_2-y_1)^2}} \). This formula computes the distance as the hypotenuse of a right triangle formed by the horizontal and vertical differences between the points.

To calculate the actual distance, as we did in the solution, we first determine the differences between the x-coordinates and y-coordinates, square them, add them together, and then find the square root of the sum. The result gives us the length of the hypotenuse, which is the direct distance between the two points, rounded to the required precision.