Problem 50 - Algebra 2

Question

Find the distance between each pair of points with the given coordinates. $$ (-3,1),(0,6) $$

Step-by-Step Solution

Answer

The distance between the points is approximately 5.83 units.

Step 1: Identify the Coordinates

We start with the given points which are \(-3, 1\) and \(0, 6\). The first point \(-3, 1\) has x-coordinate \(-3\) and y-coordinate \(1\), and the second point \(0, 6\) has x-coordinate \(0\) and y-coordinate \(6\).

Step 2: Use the Distance Formula

The formula to calculate the distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is: \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]. Substitute the given coordinates into the formula.

Step 3: Plug in the Coordinates

Substitute \(x_1 = -3\), \(y_1 = 1\), \(x_2 = 0\), and \(y_2 = 6\) into the formula: \[d = \sqrt{(0 - (-3))^2 + (6 - 1)^2}\].

Step 4: Simplify the Expression

Calculate inside the square root:\(0 - (-3)\) becomes \(3\), and \(6 - 1\) becomes \(5\). Thus, the equation becomes: \[d = \sqrt{3^2 + 5^2}\].

Step 5: Calculate the Squared Terms

Calculate \(3^2 = 9\) and \(5^2 = 25\). Substitute these values into the equation:\[d = \sqrt{9 + 25}\].

Step 6: Add the Squared Terms

Add \(9\) and \(25\) to get \(34\). Thus, the equation simplifies to:\[d = \sqrt{34}\].

Step 7: Find the Square Root

Calculate the square root of \(34\). Thus, \(d = \sqrt{34}\). Since \(\sqrt{34}\) is approximately \(5.83\), that's the distance between the points.

Key Concepts

Coordinate Geometry

In the world of mathematics, coordinate geometry is a fundamental concept that helps us navigate and understand the space around us. This branch of geometry deals primarily with points on a plane, where each point is defined by a pair of numerical values, often called coordinates. These coordinates describe the exact location of a point in a two-dimensional space using an ordered pair \(x, y\). For instance, in the exercise, the points \((-3, 1)\) and \(0, 6)\) represent locations on a grid where \(x\) is the horizontal distance from the origin and \(y\) is the vertical distance. The grid itself is formed by two axes: the x-axis (horizontal) and the y-axis (vertical). The point \(0, 0\), where these two axes intersect, is called the origin. Distances and relationships between points on this grid are often determined using key geometric principles and formulas. Coordinate geometry forms the basis for a wide range of applications, from designing computer graphics to solving problems in physics and engineering. It's the foundational tool for analyzing plain graphs, calculating areas, and much more.

Distance Between Two Points

Finding the distance between two points on a plane is a common task in coordinate geometry, and it's necessary to understand the layout of points relative to one another. We calculate this distance using a formula derived from the fundamental properties of right triangles.To find the distance for the given points \((-3, 1)\) and \(0, 6)\), we use the distance formula \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]. This formula is a direct application of the Pythagorean theorem, allowing us to compute the straight-line distance regardless of the positioning of the points on the grid. By plugging in the coordinates of our points, we can simplify the expression to calculate the distance:

• The difference in x-coordinates: \(0 - (-3) = 3\).
• The difference in y-coordinates: \(6 - 1 = 5\).

Now, by substituting these differences back into the distance formula and simplifying, we discover that the distance is \(d = \sqrt{34}\). This result captures both x and y differences, reflecting the overall change needed to move directly from one point to the other.

Pythagorean Theorem

The Pythagorean theorem is a powerful tool in geometry, particularly when working with coordinate geometry to find distances. This theorem explains the relationship between the sides of a right triangle. If you have a triangle with sides of length \(a\) and \(b\), and \(c\) is the hypotenuse, then the theorem states \[a^2 + b^2 = c^2\].In the context of finding distances between two points, the change in coordinates gives us the lengths of these sides, forming a right triangle along the grid. For the points \((-3, 1)\) and \(0, 6)\), \(3\) and \(5\) are analogous to the \(a\) and \(b\) sides of our imaginary right triangle:

• Using \(3\) for the horizontal distance (along x-axis)
• Using \(5\) for the vertical distance (along y-axis)

This framework ensures we're measuring in straight lines and therefore the real, shortest path between our points, which is the hypotenuse. Hence, when these sides are squared and summed \((3^2 + 5^2)\), the square root of their sum provides us with the direct distance, \(\sqrt{34}\), between the two points. This foundational theorem ensures accurate calculations throughout various applications in coordinate geometry and beyond.