A triangle with vertices at (2,3), (3,5), and (5,3) is transformed to a triangle with vertices at (4,6), (5,8), and (7,6). Which transformation was applied?

Question

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1 point
Translation 2 units up and 2 units right
Translation 2 units up and 2 units left
Rotation 180 degrees about the origin
Rotation 90 degrees clockwise about the origin

Answer 1

To determine the transformation that was applied to the triangle with vertices at \((2,3)\), \((3,5)\), and \((5,3)\) to arrive at the triangle with vertices at \((4,6)\), \((5,8)\), and \((7,6)\), we can analyze the changes in the coordinates of the vertices.

Let's look at the transformation step-by-step:

1. Calculating the change in coordinates for each vertex:

   • From \((2,3)\) to \((4,6)\):

     • Change in x: \(4 - 2 = 2\)
     • Change in y: \(6 - 3 = 3\)

   • From \((3,5)\) to \((5,8)\):

     • Change in x: \(5 - 3 = 2\)
     • Change in y: \(8 - 5 = 3\)

   • From \((5,3)\) to \((7,6)\):

     • Change in x: \(7 - 5 = 2\)
     • Change in y: \(6 - 3 = 3\)

2. Analyzing the transformations:

   • The x-coordinates increased by \(2\) for all points.
   • The y-coordinates increased by \(3\) for all points.

This means every point is translated by \(2\) units to the right and \(3\) units up.

However, if we relate the actual answer to the options given:

• A "Translation 2 units up and 2 units right" only accounts for the x-coordinate correct (it would be \(2\) units right), but the y-coordinate does not match the transformation we calculated.
• We calculated a \(3\) unit change in y, so there isn't a correct provided translation option.

Since this doesn't match any of the choices exactly, let's refocus on the actual transformation:

From analysis, we see that the most straightforward transformation based on the findings that all vertices are moved uniformly indicates it was a translation overall.

Thus, it appears that none of the options in the question are actually correct since the y-axis movement is also inconsistent. Based on standard transformations, it should reflect as:

Translation \(2\) units right and \(3\) units up would be the best answer if that was an option.

The closest answer based solely on movement, if constrained to the provided responses, might lead to assuming: Translation 2 units up and 2 units right is incorrect due to y-disparity but closest given that \(+2\) in x is confirmed.

Would recommend stating that the answer should reflect ultimately, considering movement "2 units up" is inaccurate but points toward translation....Correct answer appears flawed in derivative options.