Problem 19 - Algebra 2 and Trigonometry

Question

The coordinates of a point are given. a. Find the distance of the point from the origin. Express approximate distances to the nearest hundredth. b. Find the measure, to the nearest degree, of the angle in standard position whose terminal side contains the given point. \((15,0)\)

Step-by-Step Solution

Answer

The distance is 15, and the angle is 0°.

Step 1: Identify the point

The given point has coordinates \(x = 15, y = 0\). This point lies on the x-axis.

Step 2: Calculate the distance from the origin

The distance of a point \( (x, y) \) from the origin \( (0, 0) \) can be found using the distance formula \(d = \sqrt{x^2 + y^2}\). Substituting the given values: \(d = \sqrt{15^2 + 0^2} = \sqrt{225} = 15\).

Step 3: Determine the angle in standard position

In standard position, the angle is measured from the positive x-axis. Since the point \( (15, 0) \) lies directly on the x-axis, the angle is \( heta = 0^\circ\) as it coincides with the positive x-axis.

Key Concepts

Distance Formula

The distance formula is a crucial tool in geometry used to calculate the distance between two points in a plane. This formula is derived from the Pythagorean theorem and is given by: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points you want to calculate the distance between.
For example, in the given problem, we are tasked to find the distance of the point \((15, 0)\) from the origin \((0, 0)\).
The formula simplifies here since we are finding the distance to the origin, which results in:

• \(x_1 = 0\), \(y_1 = 0\)
• \(x_2 = 15\), \(y_2 = 0\)

This gives us:\[ d = \sqrt{(15 - 0)^2 + (0 - 0)^2} = \sqrt{15^2} = 15 \]
Using the distance formula helps you determine how far apart two points are, which is fundamental in many geometrical computations.

Standard Position Angle

The standard position angle is measured from the positive x-axis in a counterclockwise direction. It is a foundational concept in trigonometry and angle measurement.
When a point lies on one of the axes, determining the standard position angle becomes straightforward. In our specific case, the point \((15, 0)\) lies on the positive x-axis.
Here’s how you determine the standard position angle for different quadrants and axes:

• Positive x-axis: \(0^\circ\)
• Positive y-axis: \(90^\circ\)
• Negative x-axis: \(180^\circ\)
• Negative y-axis: \(270^\circ\)

Thus, because the point \((15, 0)\) is exactly on the x-axis, the angle is \(0^\circ\). This concept of standard position is crucial for understanding rotations and transformations in coordinate geometry.

Cartesian Coordinates

The Cartesian coordinate system is a two-dimensional plane defined by two perpendicular axes: the horizontal x-axis and the vertical y-axis. Each point on this plane is determined by a pair of numerical coordinates \((x, y)\).
These coordinates describe:

• The horizontal distance from the y-axis, indicated by the x-coordinate.
• The vertical distance from the x-axis, indicated by the y-coordinate.

In this system, any point can be precisely located and referred to, making it the standard framework for plotting and analyzing geometric figures.
In our example, the point \((15, 0)\) specifies a location 15 units away from the origin along the x-axis, with no distance along the y-axis. Cartesian coordinates are extensively used in mathematics, physics, engineering, and many other fields that require spatial representation.

Origin in Geometry

The origin in geometry is the central reference point in the Cartesian coordinate system, denoted as \((0, 0)\). It is where the x-axis and y-axis intersect, serving as a starting point for measuring distances and angles.
Understanding the origin is critical because:

• It provides a baseline for measuring distances using the distance formula.
• It is the point of reference for determining angles in standard position.
• All points are defined in relation to the origin.

In the given exercise, calculating the distance from the origin simplifies the use of the distance formula, since there is no subtraction involved among the coordinates.
Whenever working with geometry in the Cartesian plane, the origin is the key point that helps in solving numerous problems related to positioning and measurement.