Problem 51 - Introductory Algebra for College Students

Question

Find the distance between each pair of points. Express answers in simplified radical form and, if necessary, round to two decimal places. $$(-4,2) \text { and }(4,17)$$

Step-by-Step Solution

Answer

The distance between the points (-4,2) and (4,17) is 17.

Step 1: Identify the coordinates

In the problem, the pair of points given are (-4,2) and (4,17). From each pair, assign the values such that for point 1, \((x_1, y_1) = (-4,2)\) and for point 2, \((x_2, y_2) = (4,17)\).

Step 2: Substitute into the distance formula

Substitute the values into the distance formula: \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\] This results in: \[d = \sqrt{(4 - (-4))^2 + (17 - 2)^2}\].

Step 3: Simplify the expression

Simplify the expression within the square root: \[d = \sqrt{(8)^2 + (15)^2}\] which further simplifies to \[d = \sqrt{64 + 225} = \sqrt{289}\].

Step 4: Compute the square root

The final step is to calculate the square root of 289 which is 17. So, \[d = 17\] which is the distance between the two points.

Key Concepts

Coordinates Geometry

Coordinates geometry, also known as analytic geometry, involves the study of geometry using a coordinate system. This method facilitates the representation of geometric shapes and the calculation of various properties related to these shapes by using algebraic equations.

In the context of calculating the distance between two points in a plane, coordinates geometry uses the Cartesian coordinate system, which is defined by an x-axis (horizontal) and a y-axis (vertical). Each point on this plane is given as an ordered pair \(x, y\), where \(x\) represents the position along the x-axis, and \(y\) the position along the y-axis.

For example, when given the points (-4, 2) and (4, 17), each number corresponds to a position on the respective axis. Understanding how to plot these points and navigate between them using the coordinate grid is crucial in solving many real-world problems through algebra.

Radical Expression Simplification

Simplification of radical expressions is an essential algebraic process, especially when dealing with the square root function. A radical expression can often contain an entire algebraic expression under the square root symbol. Simplifying such an expression involves several steps, such as expanding powers and combining like terms, to reduce the expression to its simplest form.

Take the expression \(\sqrt{64 + 225}\) for instance. To simplify, you would first perform the addition within the radical to get \(\sqrt{289}\). Since 289 is a perfect square (being 17 squared), the simplified form of this expression is just 17. It's vital for students to recognize perfect squares and understand how to deconstruct expressions to their simplest form, as it makes further algebraic manipulation and calculation much easier.

Euclidean Distance

The concept of Euclidean distance originates from the Greek mathematician Euclid's study of geometry. It reflects the idea of 'straight-line' distance between two points in a plane, which is the most direct way to travel from one point to another.

In algebra, the Euclidean distance between two points \(x_1, y_1\) and \(x_2, y_2\) is calculated using the distance formula, which is derived from the Pythagorean theorem: \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]. This formula provides the length of the hypotenuse of a right triangle formed by the horizontal and vertical displacements from one point to the other.

Understanding how to apply this formula not only allows for distance calculations but also provides a foundation for exploring more complex geometric concepts and problems, such as those involving the properties of triangles, circles, and other shapes.