Question

Find the distance between each pair of points. Round to the nearest tenth, if necessary. \(B(0,0), C(-5,6)\)

Short answer

The distance between points B and C is approximately 7.8 units.

Step-by-step solution

Understand the Distance Formula

To find the distance between two points in a coordinate plane, we use the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points.

Identify Coordinates

Identify the coordinates of the given points. Here, point B has coordinates \((x_1, y_1) = (0, 0)\) and point C has coordinates \((x_2, y_2) = (-5, 6)\).

Plug Coordinates into the Distance Formula

Substitute the coordinates of points B and C into the distance formula:\[ d = \sqrt{((-5) - 0)^2 + (6 - 0)^2} \]

Simplify the Expression

Simplify the expression inside the square root:\[ d = \sqrt{(-5)^2 + 6^2} \]\[ d = \sqrt{25 + 36} \]\[ d = \sqrt{61} \]

Approximate the Distance

Calculate the square root of 61 and round to the nearest tenth:\[ d \approx 7.8 \]

Key concepts

Coordinate Plane

The coordinate plane is a two-dimensional surface where we can place points using pairs of numbers, known as coordinates. Each point on the plane is defined by its horizontal (x-axis) and vertical (y-axis) positions.

• The horizontal axis is called the x-axis, and it usually represents the value "x" in the coordinate pairs.
• The vertical axis is called the y-axis, corresponding to the value "y" in the pairs.
• The point where the x-axis and y-axis intersect is called the origin, which has the coordinates (0, 0).

In this system, we can precisely locate any point by using two numbers: one for the x-axis and one for the y-axis. For example, the point B with coordinates (0, 0) is located at the origin, while point C with coordinates (-5, 6) is someplace above and to the left of the origin on the coordinate plane.

Distance Between Points

The distance between two points on a coordinate plane can be calculated using the distance formula. This formula gives us the straight-line distance between two coordinates, which can be thought of as the hypotenuse of a right triangle.
The distance equation is based on the differences in both x and y coordinates:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Using the exercise as an example, the coordinates of point B are (0, 0) and for point C, (-5, 6). After substituting these into the formula:

• First, calculate the change in x: \((-5) - 0 = -5\).
• Next, calculate the change in y: \(6 - 0 = 6\).
• Then, apply these values in the formula to find the distance as \(d = \sqrt{(-5)^2 + 6^2}\).
• Simplify to find \(d = \sqrt{25 + 36} = \sqrt{61}\).

This formulation allows us to assess how far apart two points are without having to draw them physically.

Pythagorean Theorem

The Pythagorean theorem is fundamental in estimating distances and is the basis of the distance formula. It relates the sides of a right-angled triangle to its hypotenuse. According to the theorem:
\[ a^2 + b^2 = c^2 \]Where:

• "a" and "b" are the lengths of the two shorter sides – known as "legs".
• "c" represents the hypotenuse, the longest side of the triangle, opposite the right angle.
This theorem is not just for triangles; it’s exceptionally useful on the coordinate plane. Whenever you calculate the distance between two points, you essentially form a right-angled triangle, with the direct path between points as the hypotenuse.
In our example, the changes in x and y (\(-5\) and \(6\)) become the legs of the triangle. By applying the Pythagorean theorem:

• Calculate \( (-5)^2 + 6^2 = 25 + 36 = 61. \)
• The distance, representing the hypotenuse, is then found by taking the square root of 61, which approximates to \(7.8\). This is the straight-line distance between the two points.
Understanding this connection helps visualize the relationship between algebra and geometry, making these concepts more intuitive.