Triangle Congruence Analysis Is Triangle ABC Congruent To A'B'C'

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Triangle congruence is a fundamental concept in geometry, ensuring that two triangles have the same shape and size. This means all corresponding sides and angles are equal. To determine if triangles are congruent, we often use transformations such as translations, rotations, reflections, and dilations. This article delves into a specific problem involving triangle ABC and its image A'B'C' after a transformation, exploring the underlying principles of congruence and the transformations that preserve it. We will analyze the given coordinates of the triangle vertices, calculate side lengths, and apply congruence criteria to reach a conclusive answer. Understanding triangle congruence is crucial not only for academic success in geometry but also for various real-world applications in fields like engineering, architecture, and computer graphics. By mastering these concepts, students can develop strong problem-solving skills and a deeper appreciation for the elegance and precision of mathematical reasoning. Let’s embark on this geometric journey to unravel the mystery of triangle congruence and discover the transformations that uphold this essential property.

Problem Statement Analyzing Triangle Mappings

The problem presented involves triangle ABC and its transformed image A'B'C'. The coordinates of the vertices are given as follows:

  • A(-5, -3)
  • B(-2, -3)
  • C(-2, -8)
  • A'(3, -5)
  • B'(3, -2)
  • C'(8, -2)

The question at hand is to determine whether triangle ABC is congruent to triangle A'B'C'. To answer this, we need to investigate the transformation that maps ABC onto A'B'C' and check if this transformation preserves congruence. Transformations that preserve congruence are known as isometries, which include translations, rotations, and reflections. Dilations, on the other hand, change the size of the figure and thus do not preserve congruence. Our approach will involve calculating the side lengths of both triangles using the distance formula and then comparing these lengths to see if they are equal. If the corresponding sides are equal in length, we can conclude that the triangles are congruent based on the Side-Side-Side (SSS) congruence criterion. Additionally, we will examine the transformation itself to determine if it is an isometry. By combining these methods, we can confidently determine whether triangle ABC is congruent to triangle A'B'C' and provide a clear explanation supporting our conclusion. This analysis will not only answer the specific question but also reinforce the understanding of congruence and transformations in geometry.

Calculating Side Lengths Applying the Distance Formula

To determine if triangle ABC is congruent to triangle A'B'C', we first need to calculate the lengths of their sides. The distance formula is the perfect tool for this, as it allows us to find the distance between two points in a coordinate plane. The distance formula is derived from the Pythagorean theorem and is given by:

d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

Where (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) are the coordinates of the two points. We will apply this formula to find the lengths of the sides of both triangles.

Side Lengths of Triangle ABC

  1. Length of AB:

    Using points A(-5, -3) and B(-2, -3):

    AB=(−2−(−5))2+(−3−(−3))2=(3)2+(0)2=9=3AB = \sqrt{(-2 - (-5))^2 + (-3 - (-3))^2} = \sqrt{(3)^2 + (0)^2} = \sqrt{9} = 3

  2. Length of BC:

    Using points B(-2, -3) and C(-2, -8):

    BC=(−2−(−2))2+(−8−(−3))2=(0)2+(−5)2=25=5BC = \sqrt{(-2 - (-2))^2 + (-8 - (-3))^2} = \sqrt{(0)^2 + (-5)^2} = \sqrt{25} = 5

  3. Length of CA:

    Using points C(-2, -8) and A(-5, -3):

    CA=(−5−(−2))2+(−3−(−8))2=(−3)2+(5)2=9+25=34CA = \sqrt{(-5 - (-2))^2 + (-3 - (-8))^2} = \sqrt{(-3)^2 + (5)^2} = \sqrt{9 + 25} = \sqrt{34}

Side Lengths of Triangle A'B'C'

  1. Length of A'B':

    Using points A'(3, -5) and B'(3, -2):

    A′B′=(3−3)2+(−2−(−5))2=(0)2+(3)2=9=3A'B' = \sqrt{(3 - 3)^2 + (-2 - (-5))^2} = \sqrt{(0)^2 + (3)^2} = \sqrt{9} = 3

  2. Length of B'C':

    Using points B'(3, -2) and C'(8, -2):

    B′C′=(8−3)2+(−2−(−2))2=(5)2+(0)2=25=5B'C' = \sqrt{(8 - 3)^2 + (-2 - (-2))^2} = \sqrt{(5)^2 + (0)^2} = \sqrt{25} = 5

  3. Length of C'A':

    Using points C'(8, -2) and A'(3, -5):

    C′A′=(3−8)2+(−5−(−2))2=(−5)2+(−3)2=25+9=34C'A' = \sqrt{(3 - 8)^2 + (-5 - (-2))^2} = \sqrt{(-5)^2 + (-3)^2} = \sqrt{25 + 9} = \sqrt{34}

Applying Congruence Criteria Side-Side-Side (SSS) Criterion

After calculating the side lengths of both triangles, we can now compare them to determine if the triangles are congruent. We have the following side lengths:

  • Triangle ABC: AB = 3, BC = 5, CA = √34
  • Triangle A'B'C': A'B' = 3, B'C' = 5, C'A' = √34

By comparing the corresponding side lengths, we observe that:

  • AB = A'B' = 3
  • BC = B'C' = 5
  • CA = C'A' = √34

Since all three pairs of corresponding sides are equal in length, we can conclude that triangle ABC is congruent to triangle A'B'C' based on the Side-Side-Side (SSS) congruence criterion. The SSS criterion states that if three sides of one triangle are congruent to the corresponding three sides of another triangle, then the two triangles are congruent. This criterion is a fundamental concept in geometry and provides a straightforward way to establish triangle congruence when side lengths are known. In this case, the equal side lengths confirm that the triangles have the same shape and size, thus satisfying the conditions for congruence. This result not only answers the question but also reinforces the practical application of congruence criteria in geometric problem-solving. The SSS criterion is particularly useful in situations where angle measurements are not readily available, making it a valuable tool in a variety of geometric contexts.

Analyzing the Transformation Identifying the Mapping

To gain a deeper understanding of why triangle ABC is congruent to triangle A'B'C', it is beneficial to analyze the transformation that maps one triangle onto the other. By examining the coordinates of the vertices before and after the transformation, we can identify the type of transformation that occurred. The coordinates are:

  • A(-5, -3) → A'(3, -5)
  • B(-2, -3) → B'(3, -2)
  • C(-2, -8) → C'(8, -2)

Let's analyze how the coordinates change. The x-coordinate of A increases by 8 units (-5 to 3), while the y-coordinate decreases by 2 units (-3 to -5). Similarly, the x-coordinate of B increases by 5 units (-2 to 3), and the y-coordinate increases by 1 unit (-3 to -2). Lastly, the x-coordinate of C increases by 10 units (-2 to 8), and the y-coordinate increases by 6 units (-8 to -2). These changes suggest that the transformation is not a simple translation, rotation, or reflection, as those transformations would result in a more consistent pattern of coordinate changes.

The transformation appears to involve a combination of a translation and a reflection or a more complex transformation. However, the critical point is that the side lengths remain the same after the transformation, as we calculated earlier. This indicates that the transformation is a rigid transformation (also known as an isometry), which preserves distances and angles. Isometries include translations, rotations, reflections, and glide reflections. Since the triangles are congruent, the transformation must be an isometry. The specific type of isometry can be determined by further analysis, but the congruence is already established by the SSS criterion. Understanding the nature of the transformation provides additional insight into the relationship between the triangles and reinforces the concept that congruence is preserved under rigid transformations.

Conclusion Final Determination of Congruence

In conclusion, we have thoroughly investigated the congruence of triangle ABC and its image A'B'C' after a transformation. By meticulously calculating the side lengths of both triangles using the distance formula, we found that:

  • AB = A'B' = 3
  • BC = B'C' = 5
  • CA = C'A' = √34

These equal corresponding side lengths unequivocally demonstrate that triangle ABC is congruent to triangle A'B'C' based on the Side-Side-Side (SSS) congruence criterion. The SSS criterion is a cornerstone of geometric proofs, providing a direct method to establish congruence when all three sides of one triangle are congruent to the corresponding sides of another triangle. Furthermore, we analyzed the transformation that maps triangle ABC onto A'B'C' and determined that it is a rigid transformation, which preserves distances and angles, further supporting our conclusion of congruence.

Therefore, the definitive answer to the question is that triangle ABC is congruent to triangle A'B'C' because all three sides of triangle ABC are equal in length to the corresponding three sides of triangle A'B'C', satisfying the SSS congruence criterion. This comprehensive analysis not only answers the specific problem but also reinforces the fundamental principles of triangle congruence and transformations in geometry. The understanding of congruence criteria and the ability to apply them effectively are crucial skills in mathematics, enabling students to solve complex geometric problems and appreciate the inherent symmetry and order within geometric figures.

Original Question:

Question 6 (Multiple Choice Worth 3 Points) (02.07 MC) Triangle ABC maps to A'B'C' as shown. A(-5;-3), B(-2;-3) C(-2;-8) A'(3;-5) B'(3;-2) C'(8;-2) Is triangle ABC congruent to triangle A'B'C'? Why, or why not?

Rewritten Question:

Given triangle ABC with vertices A(-5, -3), B(-2, -3), and C(-2, -8), and its transformed image A'B'C' with vertices A'(3, -5), B'(3, -2), and C'(8, -2), determine if triangle ABC is congruent to triangle A'B'C'. Justify your answer by providing a clear explanation based on geometric principles and congruence criteria.