Problem 19 - Algebra 1

Question

Find the distance between the two points. Round the result to the nearest hundredth if necessary. $$(-6,-2),(-3,-5)$$

Step-by-Step Solution

Answer

The distance between the points (-6, -2) and (-3, -5) is \(\sqrt{18}\) or approximately 4.24 when rounded to the nearest hundredth.

Step 1: Identify the coordinates of the two points

The given points are (-6, -2) and (-3, -5). From the given points, we have \(x_1 = -6\), \(y_1 = -2\), \(x_2 = -3\), and \(y_2 = -5\).

Step 2: Substitute these coordinates into the distance formula

Plugging these values into the formula \(d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\) to find the distance, we get \(d = \sqrt{(-3 -- 6)^2 + (-5 --2)^2}\).

Step 3: Simplify the expression under the square root

Simplify the above expression to get \(d = \sqrt{(3)^2 + (-3)^2}\).

Step 4: Continue simplifying

The above simplifies to \(d = \sqrt{9 + 9}\).

Step 5: Final Simplification

Finally, this simplifies to \(d = \sqrt{18}\), so the distance between the points (-6, -2) and (-3, -5) is \(\sqrt{18}\).

Key Concepts

Euclidean distance

The Euclidean distance is a way of measuring the true straight-line distance between two points in Euclidean space. Imagine this space as a flat surface like a coordinate plane - the way we usually represent lines and points in math.
The concept comes from the Greek mathematician Euclid, who explored geometry in this logical and structured way.
To find the Euclidean distance, we use a formula. For any two points \(x_1, y_1\) and \(x_2, y_2\) on a plane, the distance \(d\) is given by:

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

This formula might look complicated, but it's just a way to find out how far apart two points are when you draw a straight line between them. The formula calculates the horizontal and vertical distances between the points, squares these values, then adds them together before taking the square root.
In this step-by-step solution, plugging the given points from the problem into this formula helped find the distance separating them.

Coordinate Geometry

Coordinate Geometry is a branch of geometry where positions of points on a plane are described using coordinates.
These coordinates are pairs of numbers like \(x, y\), which tell us the point's position along the horizontal (x-axis) and vertical (y-axis) lines.
By using coordinates, we can easily visualize and solve geometric problems in two-dimensional space.
This approach connects plain numerical data to geometric figures, allowing for easy calculations and deeper understanding of shapes.
In the typical coordinate plane, we start from a central point called the origin (0,0) and measure how far left or right and how far up or down a point lies from this origin.
Finding the distance between points like \((-6, -2)\) and \((-3, -5)\) becomes simpler with Coordinate Geometry because it transforms this math problem into straightforward operations on numbers.

Pythagorean Theorem

The Pythagorean Theorem is an essential mathematical principle for understanding right-angled triangles.
It states that in any 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.
This can be written as \(a^2 + b^2 = c^2\). Here, \(c\) is the hypotenuse, and \(a\) and \(b\) are the other two sides.

• This theorem is beautifully simple yet powerful and is widely applied in many areas of math and science.

The distance formula for two points on a plane is actually derived from the Pythagorean Theorem.
When calculating the distance between points such as \((-6, -2)\) and \((-3, -5)\), you are effectively finding the hypotenuse of a right triangle.
Each step: subtracting x-coordinates and y-coordinates, squaring them, adding them, and finally taking the square root, replicates the process described by the Pythagorean Theorem.