Triangle XYZ Dilation Scale Factor And Trigonometry Ratios

In the realm of geometry, understanding the properties of shapes and their transformations is crucial. One such transformation is dilation, which involves scaling a shape up or down while maintaining its original form. This concept is closely tied to the idea of similarity, where two figures have the same shape but different sizes. We will explore this relationship through the specific example of triangles XYZ and ACB, where triangle XYZ has been dilated by a scale factor of 2 to create triangle ACB. This means that every side of triangle ACB is twice the length of its corresponding side in triangle XYZ, while the angles remain unchanged. The given information states that XY = 2.5 and ZX = 5.59, which are two sides of the original triangle XYZ. Additionally, we know that angles Y and C both measure 90 degrees, indicating that both triangles are right-angled triangles. Crucially, angles A and X are congruent, meaning they have the same measure. This congruence of angles, combined with the right angles, provides a foundation for further analysis using trigonometric ratios.

To fully grasp the implications of this dilation, we must delve into the properties that are preserved and those that change. While side lengths are directly affected by the scale factor, angle measures remain invariant. This preservation of angles is a cornerstone of similarity, ensuring that the two triangles maintain the same shape. This leads us to the heart of the problem: determining the cosine of angle X (cos ∠X) and comparing it to the cosine of angle A (cos ∠A). Given that angles A and X are congruent, their trigonometric ratios, including cosine, will be equal. Therefore, calculating cos ∠X will automatically provide us with cos ∠A. The challenge then lies in utilizing the given side lengths of triangle XYZ to compute this cosine value. This involves recalling the definition of cosine in a right-angled triangle, which is the ratio of the adjacent side to the hypotenuse. By identifying the sides adjacent to angle X and the hypotenuse in triangle XYZ, we can accurately calculate cos ∠X and, by extension, cos ∠A. This exploration will not only demonstrate the application of trigonometric principles but also highlight the profound connection between dilation, similarity, and trigonometric ratios.

The significance of this problem extends beyond the immediate calculations. It underscores the power of geometric transformations in preserving certain properties while altering others. Dilation, in particular, serves as a bridge between figures of different sizes, allowing us to analyze their relationships through the lens of similarity. Understanding how trigonometric ratios behave under these transformations is essential for solving a wide range of geometric problems. For instance, in fields like architecture and engineering, scaling designs while maintaining structural integrity is a critical consideration. The principles of dilation and similarity, combined with trigonometric tools, provide the necessary framework for achieving this. Moreover, this problem offers a valuable opportunity to reinforce fundamental trigonometric concepts, such as the definition of cosine and its application in right-angled triangles. By working through this example, students can solidify their understanding of these principles and develop their problem-solving skills in geometry. The comparison of cos ∠X and cos ∠A further emphasizes the relationship between congruent angles and their trigonometric ratios, a key concept in trigonometry. This comprehensive approach ensures a thorough understanding of the interplay between geometric transformations and trigonometric functions.

Part A: Finding cos ∠X and Comparing it to cos ∠A

To find cos ∠X, we must first recall the definition of the cosine function in the context of a right-angled triangle. Cosine is defined as the ratio of the length of the side adjacent to the angle to the length of the hypotenuse. In triangle XYZ, angle Y is the right angle, making ZX the hypotenuse. The side adjacent to angle X is XY. We are given that XY = 2.5 and ZX = 5.59. Therefore, we can calculate cos ∠X as follows:

cos ∠X = (Adjacent Side) / (Hypotenuse) = XY / ZX = 2.5 / 5.59 ≈ 0.447

This calculation gives us the value of cos ∠X, which is approximately 0.447. Now, let's compare cos ∠X to cos ∠A. We are told that angles A and X are congruent. A fundamental property of congruent angles is that their trigonometric ratios are equal. This means that if ∠A and ∠X are congruent, then cos ∠A = cos ∠X. This equality stems directly from the definition of trigonometric functions, which are based on the ratios of sides in a triangle relative to a specific angle. Since the angles are identical, the ratios of their corresponding sides will also be identical, regardless of the size of the triangle.

The explanation for why cos ∠X compares to cos ∠A in this way lies in the properties of similar triangles and trigonometric functions. As mentioned earlier, triangle ACB is a dilation of triangle XYZ. Dilation preserves angles, which is why ∠A and ∠X are congruent. Trigonometric functions, such as cosine, are defined solely based on the angles within a triangle and the ratios of its sides. Therefore, if the angles are the same, the cosine values will be the same, regardless of the scale of the triangle. This principle is crucial in various applications, including surveying, navigation, and engineering, where scaled drawings and models are used extensively. Understanding that angles and their trigonometric ratios remain consistent under dilation allows for accurate calculations and predictions across different scales. The fact that cos ∠A equals cos ∠X directly reflects the relationship between angle congruence and trigonometric equality, a cornerstone of trigonometric analysis.

Moreover, this comparison highlights the power of trigonometric ratios in characterizing angles independently of the size of the triangle. Whether we are dealing with a small triangle XYZ or its dilated counterpart ACB, the cosine of a particular angle remains constant as long as the angle itself is unchanged. This property is what makes trigonometric ratios so valuable in various fields, from construction to computer graphics. In construction, for example, architects and engineers rely on trigonometric principles to ensure that angles in a building's design are accurately maintained during construction, regardless of the overall dimensions of the structure. Similarly, in computer graphics, trigonometric functions are used to create realistic images by accurately representing angles and proportions in scaled virtual environments. The consistent behavior of cosine and other trigonometric functions under dilation is therefore a fundamental aspect of many practical applications. By understanding this principle, we can effectively use trigonometry to solve problems involving scaled figures and transformations.

Part B: Finding the Length of AC

To find the length of AC, we can use the scale factor of the dilation. We know that triangle ACB was created by dilating triangle XYZ by a scale factor of 2. This means that every side of triangle ACB is twice the length of its corresponding side in triangle XYZ. AC corresponds to ZX in triangle XYZ. We are given that ZX = 5.59. Therefore, to find the length of AC, we simply multiply the length of ZX by the scale factor:

AC = Scale Factor * ZX = 2 * 5.59 = 11.18

Thus, the length of AC is 11.18 units. This calculation directly applies the definition of dilation, which involves scaling the dimensions of a figure by a constant factor. In this case, the scale factor of 2 indicates that the dilated triangle ACB is twice as large as the original triangle XYZ in each dimension. The direct proportionality between corresponding sides in similar figures allows us to easily determine the length of AC once we know the length of its counterpart, ZX, and the scale factor. This principle is widely used in various applications, such as mapmaking and model building, where scaling is essential for representing real-world objects and distances accurately.

The simplicity of this calculation belies its importance in understanding the properties of dilations and similar figures. It emphasizes that dilation is a uniform scaling transformation, meaning that all dimensions are scaled by the same factor. This uniformity preserves the shape of the figure while changing its size. This is why triangles XYZ and ACB are similar – they have the same angles but different side lengths. The relationship AC = 2 * ZX is a direct consequence of this similarity and the definition of dilation. It also illustrates the power of scale factors in geometric transformations. By knowing the scale factor, we can easily determine the dimensions of a dilated figure, provided we know the dimensions of the original figure. This is a fundamental concept in geometry that has numerous practical applications. Understanding how to use scale factors to calculate lengths in dilated figures is essential for problem-solving in geometry and related fields.

Furthermore, the determination of AC's length through the scale factor highlights the relationship between side lengths in similar triangles. Since triangles XYZ and ACB are similar, the ratios of their corresponding sides are equal. This means that XY/AC = YZ/CB = ZX/AC = 1/2, which is the reciprocal of the scale factor. This proportionality of sides is a key characteristic of similar figures and is often used to solve for unknown side lengths or to prove similarity between triangles. In this particular case, we used the ratio ZX/AC to find AC, but we could also have used other corresponding sides if their lengths were known. The ability to use these proportional relationships effectively is a cornerstone of geometric reasoning and problem-solving. By understanding the interplay between dilation, scale factors, and similar triangles, we gain a deeper appreciation for the beauty and power of geometric transformations.

In summary, this problem provides a comprehensive exploration of dilation, trigonometric ratios, and similar triangles. By finding cos ∠X and comparing it to cos ∠A, we reinforced the principle that congruent angles have equal trigonometric ratios. Determining the length of AC demonstrated the direct application of the scale factor in dilation. These exercises underscore the fundamental relationships between geometric transformations, angle congruence, side length proportionality, and trigonometric functions. A thorough understanding of these concepts is essential for success in geometry and its applications in various fields.