Question

which triangle is a translation of triangle x?
triangle a
triangle b
triangle c
triangle d

Explanation:

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<reasoning>

Identify properties of triangle X

We first analyze the original triangle \(X\).

• It is a right-angled triangle.
• The right angle is located at the vertex \((1, 2)\).
• The vertical leg goes from \((1, 2)\) to \((1, 5)\), which has a length of \(3\) units.
• The horizontal leg goes from \((1, 2)\) to \((3, 2)\), which has a length of \(2\) units.
• The orientation has the vertical leg on the left and the horizontal leg on the bottom.

Analyze translation properties

Using the Translation Rule and Coordinate Transformations concepts, a translation slides a figure in any direction without rotating, reflecting, or changing its size.

• The translated triangle must be congruent to triangle \(X\).
• It must have a vertical leg of length \(3\) and a horizontal leg of length \(2\).
• It must maintain the exact same orientation (the right angle must be at the bottom-left of the triangle).

Evaluate the options

Let's check each candidate triangle:

• Triangle A:
• The right angle is at \((-4, -4)\).
• The vertical leg goes from \((-4, -4)\) to \((-4, -2)\), length = \(2\).
• The horizontal leg goes from \((-4, -4)\) to \((-1, -4)\), length = \(3\).
• Since the dimensions are changed (rotated/reflected/resized), this is not a translation.

• Triangle B:
• The right angle is at \((-3, -1)\).
• The vertical leg goes from \((-3, -1)\) to \((-3, 2)\), length = \(3\).
• The horizontal leg goes from \((-3, -1)\) to \((-1, -1)\), length = \(2\).
• The right angle is at the bottom-left, matching the orientation of \(X\).
• Let's check the translation vector from \(X\) to \(B\):
• Right-angle vertex of \(X\): \((1, 2)\)
• Right-angle vertex of \(B\): \((-3, -1)\)
• Translation: \((x, y)

ightarrow (x - 4, y - 3)\).

• Applying this to all vertices of \(X\):
• \((1, 2)

ightarrow (-3, -1)\)

• \((1, 5)

ightarrow (-3, 2)\)

• \((3, 2)

ightarrow (-1, -1)\)

• These match the vertices of triangle \(B\) perfectly.

• Triangle C:
• The right angle is at \((2, -5)\).
• The vertical leg has length \(3\) and horizontal leg has length \(3\). This is not congruent to \(X\).

• Triangle D:
• The right angle is at \((-5, 3)\).
• The vertical leg has length \(2\) and horizontal leg has length \(4\). This is not congruent to \(X\).

Conclusion

Triangle B is the only triangle that is a direct translation of triangle X.
</reasoning>

<answer>
<mcq-option>triangle A</mcq-option>
<mcq-correct>triangle B</mcq-correct>
<mcq-option>triangle C</mcq-option>
<mcq-option>triangle D</mcq-option>
</answer>

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Answer:

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<reasoning>

Identify properties of triangle X

We first analyze the original triangle \(X\).

• It is a right-angled triangle.
• The right angle is located at the vertex \((1, 2)\).
• The vertical leg goes from \((1, 2)\) to \((1, 5)\), which has a length of \(3\) units.
• The horizontal leg goes from \((1, 2)\) to \((3, 2)\), which has a length of \(2\) units.
• The orientation has the vertical leg on the left and the horizontal leg on the bottom.

Analyze translation properties

Using the Translation Rule and Coordinate Transformations concepts, a translation slides a figure in any direction without rotating, reflecting, or changing its size.

• The translated triangle must be congruent to triangle \(X\).
• It must have a vertical leg of length \(3\) and a horizontal leg of length \(2\).
• It must maintain the exact same orientation (the right angle must be at the bottom-left of the triangle).

Evaluate the options

Let's check each candidate triangle:

• Triangle A:
• The right angle is at \((-4, -4)\).
• The vertical leg goes from \((-4, -4)\) to \((-4, -2)\), length = \(2\).
• The horizontal leg goes from \((-4, -4)\) to \((-1, -4)\), length = \(3\).
• Since the dimensions are changed (rotated/reflected/resized), this is not a translation.

• Triangle B:
• The right angle is at \((-3, -1)\).
• The vertical leg goes from \((-3, -1)\) to \((-3, 2)\), length = \(3\).
• The horizontal leg goes from \((-3, -1)\) to \((-1, -1)\), length = \(2\).
• The right angle is at the bottom-left, matching the orientation of \(X\).
• Let's check the translation vector from \(X\) to \(B\):
• Right-angle vertex of \(X\): \((1, 2)\)
• Right-angle vertex of \(B\): \((-3, -1)\)
• Translation: \((x, y)

ightarrow (x - 4, y - 3)\).

• Applying this to all vertices of \(X\):
• \((1, 2)

ightarrow (-3, -1)\)

• \((1, 5)

ightarrow (-3, 2)\)

• \((3, 2)

ightarrow (-1, -1)\)

• These match the vertices of triangle \(B\) perfectly.

• Triangle C:
• The right angle is at \((2, -5)\).
• The vertical leg has length \(3\) and horizontal leg has length \(3\). This is not congruent to \(X\).

• Triangle D:
• The right angle is at \((-5, 3)\).
• The vertical leg has length \(2\) and horizontal leg has length \(4\). This is not congruent to \(X\).

Conclusion

Triangle B is the only triangle that is a direct translation of triangle X.
</reasoning>

<answer>
<mcq-option>triangle A</mcq-option>
<mcq-correct>triangle B</mcq-correct>
<mcq-option>triangle C</mcq-option>
<mcq-option>triangle D</mcq-option>
</answer>

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