Question

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

Explanation:

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<reasoning>

Identify the properties of triangle P

Using the Coordinate Transformations knowledge point
\[

$$\begin{aligned} &\text{Vertices of } P: (-4, -2), (-4, -5), (-1, -5)\\ &\text{Orientation: Right angle at } (-4, -5)\\ &\text{Vertical leg length: } 3 \text{ units}\\ &\text{Horizontal leg length: } 3 \text{ units} \end{aligned}$$

\]

Analyze the candidate triangles

Using the Translation Rule knowledge point
\[

$$\begin{aligned} &\text{A translation preserves orientation, size, and shape (rigid motion without rotation/reflection).}\\ &\text{Triangle A: Isosceles triangle (not a right triangle).}\\ &\text{Triangle B: Vertices } (1, -2), (1, -5), (5, -5). \text{ Horizontal leg length is } 4 e 3.\\ &\text{Triangle C: Vertices } (1, 5), (1, 2), (4, 2). \text{ Right angle at } (1, 2). \text{ Legs: } 3 \text{ and } 3. \text{ Same orientation as } P.\\ &\text{Triangle D: Vertices } (-5, 5), (-5, 3), (-1, 3). \text{ Horizontal leg length is } 4 e 3. \end{aligned}$$

\]

Verify the translation vector for triangle C

Using the Translation Rule knowledge point
\[

$$\begin{aligned} &(x, y) \to (x + 5, y + 7)\\ &(-4, -2) \to (1, 5)\\ &(-4, -5) \to (1, 2)\\ &(-1, -5) \to (4, 2) \end{aligned}$$

\]
</reasoning>

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

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Answer:

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<reasoning>

Identify the properties of triangle P

Using the Coordinate Transformations knowledge point
\[

$$\begin{aligned} &\text{Vertices of } P: (-4, -2), (-4, -5), (-1, -5)\\ &\text{Orientation: Right angle at } (-4, -5)\\ &\text{Vertical leg length: } 3 \text{ units}\\ &\text{Horizontal leg length: } 3 \text{ units} \end{aligned}$$

\]

Analyze the candidate triangles

Using the Translation Rule knowledge point
\[

$$\begin{aligned} &\text{A translation preserves orientation, size, and shape (rigid motion without rotation/reflection).}\\ &\text{Triangle A: Isosceles triangle (not a right triangle).}\\ &\text{Triangle B: Vertices } (1, -2), (1, -5), (5, -5). \text{ Horizontal leg length is } 4 e 3.\\ &\text{Triangle C: Vertices } (1, 5), (1, 2), (4, 2). \text{ Right angle at } (1, 2). \text{ Legs: } 3 \text{ and } 3. \text{ Same orientation as } P.\\ &\text{Triangle D: Vertices } (-5, 5), (-5, 3), (-1, 3). \text{ Horizontal leg length is } 4 e 3. \end{aligned}$$

\]

Verify the translation vector for triangle C

Using the Translation Rule knowledge point
\[

$$\begin{aligned} &(x, y) \to (x + 5, y + 7)\\ &(-4, -2) \to (1, 5)\\ &(-4, -5) \to (1, 2)\\ &(-1, -5) \to (4, 2) \end{aligned}$$

\]
</reasoning>

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

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