Problem 9 - Algebra 2

Question

The map of a mall is overlaid with a numeric grid. The kiosk for the cell phone store is halfway between Terry's Ice Cream and the See Clearly eyeglass store. If the ice cream store is at \((2,4)\) and the eyeglass store is at \((78,46),\) find the distance the kiosk is from the eyeglass store.

Step-by-Step Solution

Answer

The kiosk is approximately 43.41 units from the eyeglass store.

Step 1: Understand the Problem

We need to find out how far the kiosk is from the See Clearly eyeglass store. The kiosk is located halfway between Terry's Ice Cream and the eyeglass store, based on given coordinates: (2,4) for the ice cream store and (78,46) for the eyeglass store.

Step 2: Find the Midpoint Between the Stores

The midpoint, or half-way point, between two points \(x_1, y_1\) and \(x_2, y_2\) is calculated with the formula: \(\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\). So, the midpoint is \((\frac{2+78}{2}, \frac{4+46}{2})\) = (40, 25). Thus, the kiosk is at \(40, 25\).

Step 3: Calculate the Distance from the Kiosk to the Eyeglass Store

The Euclidean distance between two points \(x_1, y_1\) and \(x_2, y_2\) is given by the distance formula: \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). Plugging in the coordinates of the kiosk (40, 25) and the eyeglass store (78, 46), we have: \(\sqrt{(78 - 40)^2 + (46 - 25)^2}\) which simplifies to \(\sqrt{38^2 + 21^2} = \sqrt{1444 + 441} = \sqrt{1885}\).

Step 4: Compute the Square Root

Now, calculate the square root of 1885 to find the actual distance. The square root of 1885 is approximately 43.41.

Key Concepts

Distance Formula

The Distance Formula is a powerful tool in geometry. It helps us find how far apart two points are on a grid. This is key when dealing with maps or any layout that uses coordinates. The formula for finding the distance between two points

• The distance between coordinates \((x_1, y_1)\) and \((x_2, y_2)\) is found using: \[ \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]
• It looks at the horizontal difference \((x_2 - x_1)\) and the vertical difference \((y_2 - y_1)\) separately.

After finding these two differences, we square and sum them. Finally, the square root gives the straight-line, or direct, distance between the points. In our problem, this helped measure the distance from the kiosk to the eye store.

Coordinate Geometry

Coordinate Geometry, often called analytical geometry, is all about finding relationships between points on a numerical grid.

• In this realm, positions are defined using ordered pairs like \((x, y)\).
• Each point has a unique spot on the grid, providing clarity about locations.

Using Coordinate Geometry helps visualize problems better. It aids in connecting algebraic solutions and geometric figures. For our exercise, we were given coordinates of Terry's Ice Cream and See Clearly eyeglass store. By using these coordinates, we determined crucial points and locations such as the kiosk's midpoint position. This interplay of algebra and geometry simplifies many complex problems into solvable equations.

Euclidean Distance

Euclidean Distance is a term derived from the Greek mathematician Euclid. It refers to the 'ordinary' distance we compute between two points in a plane.

• Think of it as the dashboard distance on a flat surface. It's the simplest way to describe how far two locations are from each other using a straight line.
• This distance treats the plane as flat, making calculations straightforward using basic Pythagorean principles.

The formula used for Euclidean Distance is the same as the Distance Formula mentioned earlier. By applying it, we found the actual distance between the kiosk and the eyeglass store to be approximately 43.41. This represents the straight-line route on our coordinate grid.