If Polygon ABCDE Is Dilated By A Scale Factor Of 3, With The Center Of Dilation At The Origin, To Create Polygon A'B'C'D'E', And The Endpoints Of BC Are B(3,5) And C(5,10), What Is The Slope Of B'C'?

Jun 18, 2025 by ADMIN 200 views

Understanding Dilations and Slopes A Comprehensive Guide to Polygon Transformations

In the realm of geometry, understanding transformations is crucial for grasping the fundamental properties of shapes and their relationships. Among these transformations, dilation stands out as a core concept, involving the scaling of a figure about a fixed point. This article delves into the intricacies of dilations, specifically focusing on how they affect the coordinates and slopes of geometric figures. We will explore a detailed example involving a polygon dilated by a scale factor, providing a step-by-step explanation to calculate the slope of a transformed line segment. This comprehensive guide is designed to enhance your understanding of dilations and their practical applications in coordinate geometry.

Dilation is a transformation that alters the size of a figure without changing its shape. It is characterized by two essential components: the center of dilation and the scale factor. The center of dilation is the fixed point about which the figure is scaled, while the scale factor determines the extent of the enlargement or reduction. If the scale factor is greater than 1, the figure is enlarged; if it is between 0 and 1, the figure is reduced. When the scale factor is exactly 1, the figure remains unchanged. Understanding these fundamentals is crucial for predicting how the coordinates of a figure will transform under dilation.

When a polygon is dilated, each of its vertices moves along a line that extends from the center of dilation through the original point. The distance of the new point from the center of dilation is the original distance multiplied by the scale factor. For instance, if a point is 2 units away from the center of dilation and the scale factor is 3, the dilated point will be 6 units away from the center. This uniform scaling ensures that the shape of the polygon remains the same, while its size changes proportionally. The concept of dilation is not only vital in theoretical geometry but also has practical applications in fields like computer graphics, where scaling and resizing objects are common operations.

In this article, we will primarily focus on dilations centered at the origin (0,0). This simplifies the transformation process, as the new coordinates of a point can be found by simply multiplying the original coordinates by the scale factor. For example, if a point (x, y) is dilated by a scale factor of k centered at the origin, the new coordinates will be (kx, ky). This straightforward relationship makes calculations more manageable and helps in visualizing the effect of dilation on the figure. However, it's important to note that dilations can also occur with centers other than the origin, which require a slightly more complex approach involving vector translations and scaling.

Consider polygon ABCDE, which is subjected to a dilation with a scale factor of 3, centered at the origin. This transformation results in a new polygon, A'B'C'D'E', where each vertex of the original polygon has been scaled away from the origin by a factor of 3. The problem specifically asks us to determine the slope of the line segment B'C', given the coordinates of the endpoints of BC as B(3,5) and C(5,10). To solve this, we must first understand how dilation affects the coordinates of the points and subsequently how these transformed coordinates influence the slope of the line segment.

The key to solving this problem lies in the properties of dilation. Since the dilation is centered at the origin, the coordinates of the new vertices B' and C' can be easily calculated by multiplying the coordinates of B and C by the scale factor. This process maintains the proportionality of the figure while altering its size. Once we have the coordinates of B' and C', we can apply the slope formula to find the slope of the line segment B'C'. This step-by-step approach ensures that we accurately determine the final answer, demonstrating the practical application of dilation principles in coordinate geometry.

Furthermore, this problem highlights an important property of dilations: they preserve the shape of the figure but not necessarily the size or position. The angles within the polygon remain unchanged, and parallel lines remain parallel after dilation. However, the lengths of the sides are scaled by the same factor as the scale of dilation. This preservation of shape is what distinguishes dilation from other transformations like reflections or rotations, which may change the orientation or shape of the figure. Understanding these properties is crucial for solving a wide range of geometry problems involving transformations.

1. Determine the Coordinates of B' and C'

First, we need to find the coordinates of the dilated points B' and C'. Given that the dilation has a scale factor of 3 and is centered at the origin, we multiply the coordinates of B(3,5) and C(5,10) by 3.

For B(3,5), the dilated point B' will have coordinates (3 * 3, 3 * 5) = (9,15).
For C(5,10), the dilated point C' will have coordinates (3 * 5, 3 * 10) = (15,30).

This process is a direct application of the definition of dilation centered at the origin, where each coordinate is scaled by the scale factor. Understanding this step is crucial, as it forms the foundation for calculating the slope of the line segment B'C'. The ability to quickly and accurately determine the coordinates of dilated points is a valuable skill in coordinate geometry, allowing for efficient problem-solving.

2. Apply the Slope Formula

Now that we have the coordinates of B'(9,15) and C'(15,30), we can calculate the slope of the line segment B'C'. The slope formula, often denoted as m, is given by:

m = (y2 - y1) / (x2 - x1)

where (x1, y1) and (x2, y2) are the coordinates of two points on the line. In our case, (x1, y1) = (9,15) and (x2, y2) = (15,30).

Substituting these values into the slope formula, we get:

m = (30 - 15) / (15 - 9) = 15 / 6

3. Simplify the Slope

The slope we calculated, 15/6, can be simplified to its lowest terms. Both 15 and 6 are divisible by 3, so we divide both the numerator and the denominator by 3:

m = (15 / 3) / (6 / 3) = 5 / 2

Thus, the slope of the line segment B'C' is 5/2. This simplified fraction represents the ratio of the vertical change (rise) to the horizontal change (run) along the line segment. The ability to simplify fractions is an essential skill in mathematics, ensuring that answers are presented in their most concise and understandable form.

The slope of B'C' is 5/2.

This problem effectively demonstrates several key concepts in geometry, particularly those related to transformations and coordinate geometry. Dilation, as a transformation, plays a central role, highlighting how figures can be scaled while maintaining their shape. The scale factor determines the extent of the enlargement or reduction, and the center of dilation acts as the fixed point around which the scaling occurs. In this case, the dilation is centered at the origin, simplifying the transformation process to a direct multiplication of coordinates by the scale factor. This understanding is crucial for solving similar problems involving dilations.

Another significant concept is the slope of a line, which measures its steepness and direction. The slope formula, m = (y2 - y1) / (x2 - x1), is a fundamental tool in coordinate geometry, allowing us to quantify the rate of change between two points on a line. The problem reinforces the application of this formula, demonstrating how it can be used to find the slope of a line segment after a transformation. Furthermore, the problem emphasizes the importance of simplifying fractions, ensuring that the final answer is presented in its most reduced form.

The relationship between dilation and slope is particularly noteworthy. Dilation, being a similarity transformation, preserves the angles of the figure and maintains the parallelism of lines. Consequently, the slope of a line segment remains unchanged under dilation. In this problem, the slope of BC is the same as the slope of B'C'. This property is a direct consequence of the geometric principles governing dilations and is a valuable insight for problem-solving. Understanding that dilations preserve slopes can simplify calculations and provide a shortcut in many geometric problems.

The concepts explored in this article have numerous practical applications beyond theoretical geometry. In computer graphics, dilation is used extensively for scaling images and objects. Architects and engineers use similar principles when creating scaled models of buildings and structures. The understanding of dilations and slopes is also essential in fields like cartography, where maps are scaled representations of geographical areas.

For further exploration, consider investigating dilations centered at points other than the origin. This involves a slightly more complex transformation process, requiring vector translations in addition to scaling. Another interesting area to explore is the relationship between dilations and other transformations, such as rotations and reflections. Combining these transformations can create complex geometric patterns and is a fundamental concept in transformational geometry. Additionally, exploring the applications of dilations in three-dimensional space can provide a deeper understanding of spatial transformations.

In conclusion, the problem of finding the slope of a dilated line segment provides a valuable exercise in applying geometric principles. By understanding the properties of dilation, the slope formula, and the preservation of slopes under dilation, you can effectively solve a wide range of problems in coordinate geometry. This knowledge not only enhances your mathematical skills but also provides a foundation for further exploration in various fields that rely on spatial transformations.