Problem 28 - Precalculus with Limits

Question

In Exercises 27-38, find the distance between the points. \( (1, 4) \), \( (8, 4) \)

Step-by-Step Solution

Answer

The distance between the points (1, 4) and (8, 4) is 7.

Step 1: Substitute the coordinates into the distance formula

The coordinates are (1,4) and (8,4) for Point1 and Point2 respectively. Substitute these values into the distance formula. Thus the formula becomes \sqrt {(8 - 1)^2 + (4 - 4)^2}.

Step 2: Calculate the squares

Calculate the square of the differences in the formula. This gives \sqrt {(7)^2 + (0)^2}, which becomes \sqrt {49 + 0}.

Step 3: Compute the square root

Finally, calculate the square root in the formula. The answer is \sqrt {49}.

Step 4: Final calculation

The square root of 49 is 7. So, the distance between the points (1, 4) and (8, 4) is 7.

Key Concepts

Distance Formula

Imagine you’re on a treasure hunt, where X marks the spot on a map. To find the distance between where you are (Point A) and the treasure (Point B), we use a mathematical 'map' called the distance formula. In coordinate geometry, this formula is like a GPS for finding the exact distance between two points on a grid.

The distance formula is expressed as \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). It’s a direct application of the Pythagorean theorem from a right triangle formed by horizontal and vertical lines connecting the two points. We calculate the changes (\(\text{also known as 'deltas'}\)) in x (horizontal) and y (vertical) coordinates, square each, sum them up, and finally extract the square root. Those straightforward steps lead us straight to the numerical 'treasure' - the distance between our two points.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This method combines algebra and geometry to describe the position of points, lines, and shapes. Think of it as a playground grid where each place has its unique point on the map, called coordinates, written as \( (x, y) \).

In our treasure hunt scenario, the points \( (1, 4) \) and \( (8, 4) \) represent definite spots on this grid. By employing coordinate geometry, we can analyze the spatial relationships between these points. When points share a common coordinate, like in our exercise, it indicates they lie along the same horizontal or vertical line - a helpful clue that simplifies our calculation of distance!

Square Root

Picture the square root as the 'undo' button for a squared number. Mathematically, it's represented by the symbol \(\sqrt{}\). When you square a number (multiply it by itself), the square root brings you back to the original number. In our exercise, we deal with the square root of 49, which breaks down into \(\sqrt{49} = 7\), since 7 multiplied by itself gives 49.

Why is the square root important in our distance treasure hunt? After we add the squares of our coordinate changes, we must find the original distance, not its square. That's where the square root comes into play, turning squared steps back into straight-line distance - and guiding us to the precise point of the treasure!