Problem 53 - 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. $$(6,-1) \text { and }(9,5)$$

Step-by-Step Solution

Answer

The distance between the points (6,-1) and (9,5) is \(3 \sqrt{5}\)

Step 1: Identify the coordinates

First, identify the coordinates of the points between which the distance is to be found. Let (6, -1) be \(P(x_1, y_1) = P(6, -1)\) and (9, 5) be \(Q(x_2, y_2) = Q(9, 5)\).

Step 2: Use the distance formula

Now plug the coordinates in the distance formula \(d = \sqrt {(x_2 - x_1)^2 + (y_2 - y_1)^2}\). Here \(x_1 = 6\), \(y_1 = -1\), \(x_2 = 9\), and \(y_2 = 5\). Therefore, \(d = \sqrt {(9 - 6)^2 + (5 - (-1))^2}\).

Step 3: Simplify the expression

Simplify the expression under the square root as follows: \(d = \sqrt {(3)^2 + (6)^2} = \sqrt {9 + 36} = \sqrt {45}\).

Step 4: Simplify further

Further simplify \(d\) as \(\sqrt {45}\) equals to \(\sqrt{9*5} = 3 \sqrt{5}\)

Step 5: Round to two decimal places if necessary

It is not necessary to round the answer in this case, as the simplified radical form does not include any decimals. However, if the question specified to do so, we would round it as follows: \(3 \sqrt{5} \approx 6.71\)

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, combines algebra and geometry to solve problems concerning positions and distances of points on a plane. It uses a coordinate system – typically Cartesian coordinates – which allows each point to be uniquely defined by a pair of numbers, representing its location on horizontal (x-axis) and vertical (y-axis) axes.

To determine the distance between any two points on a plane, the coordinates of these points are crucial. In our exercise, the points (6,−1) and (9,5) provide these coordinates. Each pair of numbers tells us how far the point is from the origin in two directions:

• The first number is the x-coordinate, indicating distance along the horizontal axis.
• The second number is the y-coordinate, showing distance along the vertical axis.

Understanding these coordinates allows students to visualize and calculate spatial relationships accurately.

Radical Simplification

Radical simplification is the process of rewriting a radical (a root expression) in its simplest form. This technique is often necessary to express solutions more clearly or reduce them to a standardized format.

In our example, we calculate the distance using the distance formula, which results in a square root: \(\sqrt{45}\). Here, simplification involves finding any perfect square factors of the integer under the radical sign. In this case, 45 can be broken down into its prime factors as \(9 \times 5\), where 9 is a perfect square. Therefore, \(\sqrt{45} = \sqrt{9 \times 5} = 3\sqrt{5}\). Simplifying radicals makes them easier to understand and mathematically manipulate for further calculations.

Additionally, expressing numbers in simplified radical form, like \(3\sqrt{5}\), is often preferred in mathematics since it provides an exact value, unlike decimal approximations.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry describing the relationship between the sides of a right triangle. It states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

Though it primarily concerns triangles, the Pythagorean Theorem is also used to derive the distance formula used in coordinate geometry. When calculating the distance between two points, we essentially form a right triangle where the distance between the points is the hypotenuse. Therefore, the distance formula is an application of this theorem: \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). Each component \((x_2 - x_1)\) and \((y_2 - y_1)\) represents the two shorter sides of the triangle, and together they help find the required distance.

This consistent use of an ancient theorem in modern problems illustrates its timelessness and importance in mathematical problem-solving, demonstrating the interconnectedness of various branches within mathematics.