To demonstrate that a triangle in quadrant IV is congruent to a triangle in quadrant II, we can analyze the transformations that have taken place to move one triangle to the other. These transformations can include translation, rotation, and reflection. Here’s a step-by-step description of possible transformations:
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Reflection: The first step is to reflect the triangle in quadrant IV over the y-axis. This will move the triangle into quadrant III. Each point of the triangle (let’s call it \( A(x, y) \)) will transform to its reflected position \( A'(-x, y) \).
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Translation: After reflection, we need to translate the triangle from quadrant III to quadrant II. This can involve moving the triangle horizontally to the left along the x-axis. For instance, if the triangle’s vertices are at points like \( A'(-x, y) \), \( B'(-x_1, y_1) \), and \( C'(-x_2, y_2) \), we could translate it to positions like \( A''(-x - d, y) \), \( B''(-x_1 - d, y_1) \), and \( C''(-x_2 - d, y_2) \), where \( d \) is a positive value that allows the triangle to move into quadrant II.
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Verification of Congruence: After completing these transformations, the final triangle in quadrant II will be congruent to the original triangle in quadrant IV. This is because both triangles have the same size and shape, as congruence is preserved under reflection and translation. The angles and side lengths of the triangles will remain unchanged throughout these transformations.
By identifying the transformations of reflection and translation, we can confidently conclude that the triangle in quadrant IV is congruent to the triangle in quadrant II.