Problem 17 - Algebra 1

Question

Find the distance between the two points. Round the result to the nearest hundredth if necessary. $$(5,8),(-2,3)$$

Step-by-Step Solution

Answer

The distance between the points (5,8) and (-2,3) is approximated as 8.60.

Step 1: Identify given coordinates

The two points given are (5,8) and (-2,3). Hence, \(x_1 = 5, y_1 = 8, x_2 = -2, y_2 = 3\).

Step 2: Input values into formula

Substitute the values of the coordinates into the formula. So the distance formula is, \(\sqrt{(-2 - 5)^2 + (3 - 8)^2}\).

Step 3: Simplifying inside the square root

First simplify the expression inside the square root. This simplifies to \(\sqrt{(-7)^2 + (-5)^2} = \sqrt{49 + 25}\).

Step 4: Calculation of distance

Add up the two results inside the square root. This leads to \(\sqrt{74}.\) In decimal form, this equals approximately 8.60.

Step 5: Round off the final result

Rounding to the nearest hundredth, the final distance becomes 8.60.

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, deals with the connection between algebra and geometric figures. This field provides a powerful way of describing and analyzing two-dimensional shapes using a set of coordinates. In particular, the distance formula is a remarkable application of coordinate geometry. It allows us to calculate the exact distance between any two points on a plane using their coordinates.

When given two points, say \(A(x_1, y_1)\) and \(B(x_2, y_2)\), the distance between them can be found by considering them as vertices of a right triangle where the horizontal and vertical distances between them are the triangle's legs. This brings us to another essential concept in coordinate geometry: the Cartesian coordinate system, which defines each point uniquely using an ordered pair of numbers.

Square Root Simplification

The square root simplification process involves reducing radical expressions to their simplest form. This is commonly seen when using the distance formula in coordinate geometry. After calculating the sum inside the square root, as we did with \(\sqrt{49 + 25}\), we aim to simplify it.

Here's what simplification might look like:

• Identify perfect squares within the radicand (the number inside the square root).
• Take the square root of the perfect squares separately from those numbers that are not perfect squares.
• Combine and simplify the separate results if possible.

In our example, we didn't have any further simplification since 74 is not a perfect square. But imagine if we had \(\sqrt{36}\), knowing that 36 is a perfect square, we could easily simplify that to '6' since \(6^2 = 36\).

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry that states the square of the hypotenuse (the side opposite the right angle) in a right-angled triangle is equal to the sum of the squares of the other two sides. This theorem can be expressed algebraically as \(a^2 + b^2 = c^2\), where \(c\) is the hypotenuse while \(a\) and \(b\) are the other two sides.

The distance formula \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\) is a direct application of this theorem in coordinate geometry, with \(d\) representing the hypotenuse, and \(x\) and \(y\) representing the difference in coordinates, analogous to the sides \(a\) and \(b\). The theorem is not only crucial for academic purposes but also has practical applications in various fields including navigation, architecture, and engineering.