Problem 21 - Contemporary Precalculus

Question

Which of the following points is closest to the origin? $$(4,4.2),(-3.5,4.6),(-3,-5),(2,-5.5)$$

Step-by-Step Solution

Answer

Answer: The point closest to the origin is (-3.5, 4.6).

Step 1: Apply the distance formula to each point

For each point, we will apply the distance formula with the origin \((0,0)\) as one of the points. This will give us the distance between each point and the origin. For point \((4,4.2)\): \(d = \sqrt{(4 - 0)^2 + (4.2 - 0)^2} = \sqrt{16 + 17.64} = \sqrt{33.64}\) For point \((-3.5,4.6)\): \(d = \sqrt{(-3.5 - 0)^2 + (4.6 - 0)^2} = \sqrt{12.25 + 21.16} = \sqrt{33.41}\) For point \((-3,-5)\): \(d = \sqrt{(-3 - 0)^2 + (-5 - 0)^2} = \sqrt{9 + 25} = \sqrt{34}\) For point \((2,-5.5)\): \(d = \sqrt{(2 - 0)^2 + (-5.5 - 0)^2} = \sqrt{4 + 30.25} = \sqrt{34.25}\)

Step 2: Compare the distances

Now we will compare the calculated distances: For point \((4,4.2)\): \(\sqrt{33.64}\) For point \((-3.5,4.6)\): \(\sqrt{33.41}\) For point \((-3,-5)\): \(\sqrt{34}\) For point \((2,-5.5)\): \(\sqrt{34.25}\) The smallest distance is \(\sqrt{33.41}\), which corresponds to the point \((-3.5, 4.6)\).

Step 3: Determine the point closest to the origin

From our comparisons, the point closest to the origin is \((-3.5, 4.6)\).

Key Concepts

Coordinate Geometry

Coordinate geometry, or analytic geometry, is a critical method in mathematics used to represent and analyze geometrical objects using coordinates and algebra. It helps us connect geometry and algebra through graphs of equations. By placing shapes in a coordinate plane in a logical manner, we can study their properties and relations more efficiently.
The coordinate plane consists of two axes:

• The x-axis which runs horizontally.
• The y-axis which runs vertically.

The intersection point of these axes is called the origin, denoted as (0,0).
A crucial tool in coordinate geometry is the distance formula, which finds the distance between two points in the coordinate system. This makes it possible to solve many geometric problems, such as finding the midpoint of a line segment or determining the area of a polygon.

Cartesian Coordinates

Cartesian coordinates are a system used in coordinate geometry to uniquely specify points on a plane by an ordered pair of numbers. Named after René Descartes, this system makes it easier to denote positions mathematically.
In this system:

• Each point is described by a pair of numbers (x, y), where 'x' is the horizontal position, and 'y' is the vertical position.
• The coordinates are based on their distance from the two perpendicular lines known as axes.
• The position of points can be easily described and plotted on a graph.

In our example, points such as (4, 4.2) or (-3.5, 4.6) are specified by their 'x' and 'y' values which come from counting units along each respective axis from the origin.

Calculating Distances

To find the distance between two points in a Cartesian coordinate system, we use the distance formula derived from the Pythagorean theorem.
For two points (x_1, y_1) and (x_2, y_2), the distance 'd' is calculated as:

• \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]

This formula calculates the straight-line distance between two points, essentially finding the hypotenuse of a right-angled triangle.
In our exercise, we've applied the distance formula to find the distance of each point from the origin (0,0). Each calculation answers the problem by determining the shortest distance, helping us identify that point (-3.5, 4.6) is closest to the origin. This practical use of the distance formula shows its versatility in answering everyday distance queries in a planar system.