Problem 3 - Algebra 1

Question

Use the coordinate plane to estimate the distance between the two points. Then use the distance formula to find the distance between the points. Round the result to the nearest hundredth. \((1,5),(-3,1)\) (GRAPH CAN'T COPY)

Step-by-Step Solution

Answer

The distance between the points \(A(1,5)\) and \(B(-3,1)\) is 5.66 units.

Step 1: Identify the Coordinates

The two points given in this problem are \(A(1,5)\) and \(B(-3,1)\). Therefore, \(A(x1,y1)\) and \(B(x2,y2)\). Specifically, \(x1 = 1, y1 = 5, x2 = -3\), and \(y2 = 1\).

Step 2: Apply the Distance Formula

Substitute the coordinates of each point into the distance formula \(\sqrt{(x2 - x1)^2 + (y2 - y1)^2}\). This gives: \(\sqrt{(-3 - 1)^2 + (1 - 5)^2}\), which simplifies to: \(\sqrt{(-4)^2 + (-4)^2} = \sqrt{16 + 16}\).

Step 3: Calculate the Distance

Continuing with the calculation, we get \(\sqrt{32}\).

Step 4: Round the Result

Converting \(\sqrt{32}\) into a decimal gives approximately 5.66. Rounding this to the nearest hundredth results in 5.66.

Key Concepts

Coordinate Plane

The coordinate plane is a two-dimensional surface on which we plot points, lines, and curves. It consists of two number lines: the x-axis (horizontal) and the y-axis (vertical), which intersect at a point called the origin.

• The origin has the coordinates (0, 0).
• Every point on the plane is defined by an ordered pair (x, y).

Understanding how to read and plot points on the coordinate plane is crucial for solving problems involving the Distance Formula. Imagine our points A(1, 5) and B(-3, 1) sitting on this plane. Point A is "1 unit" to the right of the origin and "5 units" up. Meanwhile, point B is "3 units" left and "1 unit" up from the origin. Visualizing these points can help us estimate the distance between them before calculating it precisely.

Estimation

Estimation comes in handy when you need a quick, rough approximation before delving into calculations. In math—and especially on the coordinate plane—estimation can give us an idea of the distance or difference between two points without needing exact calculations right away.

• For instance, looking at points (1,5) and (-3,1) on a graph, notice how both x-coordinates and y-coordinates differ by approximately 4 units.
• A rough mental image may suggest the points are separated by a little over 5 units.

Estimates can often be useful to immediately "sense-check" your future calculations. Comparing your estimate with the precise result later on ensures that your computed answer seems reasonable.

Rounding Off

Rounding off is the process of adjusting a number to make it simpler or more manageable to work with while retaining its significance. When precise numbers are not required—or for easier communication—rounding becomes useful.

• In our example, once the exact distance was calculated as \(\sqrt{32}\), converting this to a decimal gave us approximately 5.65685.
• Rounding this to the nearest hundredth, we focus on the second digit after the decimal point, resulting in 5.66.

Understanding rounding rules ensures you simplify numbers accurately: 1. If the digit after the place you round to is 5 or more, round up.2. Otherwise, round down.In this way, you're always able to provide a correctly rounded final result.