Transformations And Similarity Analyzing Triangles ΔXYZ And ΔX'Y'Z'

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Which statements must be true about the two triangles ΔXYZ and ΔX'Y'Z' after a reflection over a vertical line and a dilation by a scale factor of 1/2?

In the realm of geometry, transformations play a pivotal role in altering the position, size, or orientation of shapes. When dealing with triangles, understanding how transformations affect their properties is crucial. This article delves into the specific scenario where triangle ΔXYZ undergoes a reflection over a vertical line followed by a dilation with a scale factor of 1/2, resulting in triangle ΔX'Y'Z'. We will explore the implications of these transformations and determine which statements must be true about the relationship between the two triangles.

Transformations: Reflection and Dilation

To fully grasp the relationship between ΔXYZ and ΔX'Y'Z', it's essential to understand the nature of the transformations involved: reflection and dilation.

Reflection: Mirroring Across a Line

A reflection is a transformation that flips a figure over a line, known as the line of reflection. In this case, ΔXYZ is reflected over a vertical line. This means that each point in the original triangle has a corresponding point in the reflected triangle that is equidistant from the vertical line but on the opposite side. Reflections preserve the shape and size of the figure; only the orientation is changed. Think of it as creating a mirror image of the triangle. The corresponding sides and angles remain congruent, meaning they have the same measure. Therefore, reflection is an example of an isometric transformation, which maintains the original dimensions and shape.

When ΔXYZ is reflected over a vertical line, the x-coordinates of the vertices change sign while the y-coordinates remain the same. For example, if point X has coordinates (a, b), then its reflection X' will have coordinates (-a, b). This change in x-coordinates results in a mirror image across the vertical line. It is crucial to emphasize that despite this change in orientation, the fundamental characteristics of the triangle—its angles and side lengths—remain unaltered. This is a key concept in understanding why reflected figures are congruent.

Consider a scenario where ΔXYZ has vertices X(2, 3), Y(5, 1), and Z(3, 5). If we reflect this triangle over the y-axis (which is a vertical line), the new coordinates for ΔX'Y'Z' will be X'(-2, 3), Y'(-5, 1), and Z'(-3, 5). A visual comparison of these two triangles would reveal that they are mirror images of each other, maintaining the same shape and size. This illustrative example underscores the essence of reflection as a transformation that preserves congruence while altering orientation.

Dilation: Resizing with a Scale Factor

A dilation, on the other hand, is a transformation that changes the size of a figure. It involves multiplying the coordinates of each point by a scale factor. The scale factor determines whether the figure is enlarged (scale factor > 1) or reduced (0 < scale factor < 1). In our problem, ΔXYZ is dilated by a scale factor of 1/2. This indicates that the triangle will be reduced in size, specifically to half its original dimensions. Dilation is a non-isometric transformation because it alters the size of the figure.

When a figure is dilated, its shape remains the same, but its size changes proportionally. This means that the angles of the triangle remain the same, but the side lengths are scaled by the factor. For instance, if side XY has a length of 6 units, after dilation by a scale factor of 1/2, side X'Y' will have a length of 3 units. This proportional change in side lengths is crucial for maintaining the similarity between the original and dilated figures.

To provide a clearer picture, imagine ΔXYZ with vertices at X(4, 4), Y(8, 2), and Z(6, 6). Applying a dilation with a scale factor of 1/2 would result in ΔX'Y'Z' with vertices at X'(2, 2), Y'(4, 1), and Z'(3, 3). A comparison of these triangles would show that ΔX'Y'Z' is a smaller version of ΔXYZ, but both triangles maintain the same angles. This example vividly demonstrates how dilation affects size while preserving shape, which is fundamental to the concept of similarity.

Similarity: The Key Relationship

Given these transformations, the crucial question arises: What is the relationship between ΔXYZ and ΔX'Y'Z'? The key concept here is similarity. Two figures are similar if they have the same shape but not necessarily the same size. More formally, two triangles are similar if their corresponding angles are congruent and their corresponding sides are in proportion.

The reflection over a vertical line preserves the angles and side lengths (congruence), while the dilation by a scale factor of 1/2 preserves the angles but changes the side lengths proportionally. Therefore, the combination of these transformations ensures that ΔXYZ and ΔX'Y'Z' have congruent angles and proportional sides, which is the very definition of similarity. Thus, ΔXYZ ~ ΔX'Y'Z'.

When we consider the transformations applied—reflection followed by dilation—it becomes evident why similarity is preserved. The reflection, while altering the triangle's orientation, does not change its shape or size. The subsequent dilation, however, changes the size but maintains the shape by scaling all sides proportionally. This means that the ratios of corresponding sides in the two triangles are equal, a key criterion for similarity.

To illustrate this further, suppose ΔXYZ has angles measuring 60°, 70°, and 50°. After reflection and dilation, ΔX'Y'Z' will still have angles measuring 60°, 70°, and 50°. If side XY in ΔXYZ measures 8 units and the corresponding side X'Y' in ΔX'Y'Z' measures 4 units (due to the dilation by a scale factor of 1/2), then the ratio of these sides is 1/2. Similarly, if side YZ in ΔXYZ measures 10 units, side Y'Z' in ΔX'Y'Z' will measure 5 units, maintaining the same 1/2 ratio. This constant ratio across all corresponding sides confirms that the triangles are similar.

Analyzing the Statements: Which Must Be True?

Now, let's consider the specific statements that must be true about ΔXYZ and ΔX'Y'Z'. Based on our understanding of reflection, dilation, and similarity, we can identify the correct options.

  1. ΔXYZ ~ ΔX'Y'Z': This statement is true. As discussed earlier, the combination of reflection and dilation results in similar triangles. The angles are preserved, and the sides are proportional.

  2. The area of ΔX'Y'Z' is one-fourth the area of ΔXYZ: This statement is true. When a figure is dilated by a scale factor of k, its area changes by a factor of k². In this case, the scale factor is 1/2, so the area changes by a factor of (1/2)² = 1/4. Therefore, the area of ΔX'Y'Z' is one-fourth the area of ΔXYZ.

The relationship between the areas of similar figures is a direct consequence of the proportional scaling of their sides. When the sides of a triangle are reduced by a factor of 1/2, the base and height are both reduced by the same factor. The area of a triangle is given by (1/2) * base * height. If both the base and height are multiplied by 1/2, the area is multiplied by (1/2) * (1/2) = 1/4.

For instance, if ΔXYZ has a base of 8 units and a height of 6 units, its area is (1/2) * 8 * 6 = 24 square units. After dilation by a scale factor of 1/2, ΔX'Y'Z' will have a base of 4 units and a height of 3 units, making its area (1/2) * 4 * 3 = 6 square units. This confirms that the area of ΔX'Y'Z' is indeed one-fourth the area of ΔXYZ.

  1. The perimeter of ΔX'Y'Z' is one-half the perimeter of ΔXYZ: This statement is true. Since the dilation has a scale factor of 1/2, all the sides of ΔX'Y'Z' are one-half the length of the corresponding sides of ΔXYZ. The perimeter is the sum of the side lengths, so the perimeter of ΔX'Y'Z' is also one-half the perimeter of ΔXYZ.

The perimeter of a triangle is simply the sum of the lengths of its three sides. When a triangle is dilated, each side is scaled by the same factor. Therefore, if the scale factor is 1/2, each side of the dilated triangle is half the length of the corresponding side in the original triangle. Consequently, the sum of these halved lengths (the perimeter of the dilated triangle) will be half the sum of the original lengths (the perimeter of the original triangle).

To illustrate, suppose ΔXYZ has sides of lengths 6, 8, and 10 units. Its perimeter is 6 + 8 + 10 = 24 units. After dilation by a scale factor of 1/2, ΔX'Y'Z' will have sides of lengths 3, 4, and 5 units. The perimeter of ΔX'Y'Z' is 3 + 4 + 5 = 12 units, which is indeed half the perimeter of ΔXYZ.

Conclusion: Key Takeaways

In conclusion, when ΔXYZ is reflected over a vertical line and then dilated by a scale factor of 1/2, the resulting triangle ΔX'Y'Z' maintains a clear relationship with the original. The three statements that must be true are:

  • ΔXYZ ~ ΔX'Y'Z' (The triangles are similar).
  • The area of ΔX'Y'Z' is one-fourth the area of ΔXYZ.
  • The perimeter of ΔX'Y'Z' is one-half the perimeter of ΔXYZ.

Understanding these transformations and their effects on geometric figures is essential for mastering geometry. Reflections preserve congruence, while dilations create similar figures. The interplay between these transformations leads to predictable changes in size, area, and perimeter, which are crucial for problem-solving in geometry.

By grasping these fundamental concepts, students can confidently tackle a wide array of geometric problems and develop a deeper appreciation for the elegance and precision of mathematical transformations.