Problem 5 - Calculus Single Variable

Question

Let \(A=(2,3), B=(-4,7),\) and \(C=(-5,-6) .\) Calculate the distance of each of these points to each of the others.

Step-by-Step Solution

Answer

The distances are: AB = \(2\sqrt{13}\), AC = \(\sqrt{130}\), BC = \(\sqrt{170}\).

Step 1: Calculate Distance Between Points A and B

To find the distance between points A and B, we use the distance formula: \[d_{AB} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]Substitute \((x_1, y_1) = (2, 3)\) for point A and \((x_2, y_2) = (-4, 7)\) for point B:\[d_{AB} = \sqrt{((-4) - 2)^2 + (7 - 3)^2} = \sqrt{(-6)^2 + (4)^2} = \sqrt{36 + 16} = \sqrt{52} = 2\sqrt{13}\]

Step 2: Calculate Distance Between Points A and C

Now use the distance formula to calculate the distance between points A and C:\[d_{AC} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]Substitute \((x_1, y_1) = (2, 3)\) for point A and \((x_2, y_2) = (-5, -6)\) for point C:\[d_{AC} = \sqrt{((-5) - 2)^2 + ((-6) - 3)^2} = \sqrt{(-7)^2 + (-9)^2} = \sqrt{49 + 81} = \sqrt{130} = \sqrt{130}\]

Step 3: Calculate Distance Between Points B and C

Finally, use the distance formula for points B and C:\[d_{BC} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]Substitute \((x_1, y_1) = (-4, 7)\) for point B and \((x_2, y_2) = (-5, -6)\) for point C:\[d_{BC} = \sqrt{((-5) + 4)^2 + ((-6) - 7)^2} = \sqrt{(-1)^2 + (-13)^2} = \sqrt{1 + 169} = \sqrt{170} = \sqrt{170}\]

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, allows us to describe the position of points in a plane using numerical coordinates. These coordinates, typically represented as \((x, y)\), indicate how far the point is from the x-axis and y-axis. The system gives us a way to use algebraic equations to solve geometric problems, such as determining the distance between points, slope of a line, or the equation of a circle.

In the context of our exercise, we are dealing with three points: \(A = (2, 3)\), \(B = (-4, 7)\), and \(C = (-5, -6)\). Each of these points has an x-coordinate and a y-coordinate that show their location on a Cartesian plane. By understanding these coordinates, we can apply formulas to find distances and relationships between the points.

Distance Calculation

Calculating the distance between two points in coordinate geometry is straightforward once we know their coordinates. Typically, we use the distance formula, which is derived from the Pythagorean theorem. The formula looks like this:

• \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

This equation helps us find the straight-line distance (also known as the Euclidean distance) between any two points. It essentially measures how far apart the two points are in the coordinate plane.

For instance, to find the distance between points \(A\) and \(B\), substitute \((x_1, y_1) = (2, 3)\) and \((x_2, y_2) = (-4, 7)\) into the distance formula, which yields:

• \(d_{AB} = \sqrt{((-4) - 2)^2 + (7 - 3)^2} = \sqrt{52} = 2\sqrt{13}\)

Thus, applying this method to any two points helps us understand their relative spatial relation.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in mathematics, especially used in geometry to relate the lengths of the sides of a right triangle. The theorem 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 lengths of the other two sides.

Mathematically, it is written as \(a^2 + b^2 = c^2\), where \(c\) is the hypotenuse, while \(a\) and \(b\) are the other two sides.

This theorem is the foundation for the distance formula. Imagine you are constructing a right triangle from two points in the coordinate plane. The differences in the x-coordinates and y-coordinates form the two legs of a right triangle, while the distance between the points is the hypotenuse. Thus, using this relationship, the distance formula can be shown as a direct application of the Pythagorean theorem.