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Question

Part 1
The vertices of triangle symbol ABC△ABC are Aleft parenthesis 2 comma negative 1 right parenthesis(2,−1)​, Bleft parenthesis negative 2 comma negative 3 right parenthesis(−2,−3)​, and Cleft parenthesis 4 comma 3 right parenthesis(4,3). For the translation​ below, give the vertices of triangle symbol Upper A prime Upper B prime Upper C prime△A′B′C′.
Upper T Subscript left angle negative 4 comma 5 right angle Baseline left parenthesis triangle symbol ABC right parenthesisT−4,5(△ABC)
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Part 1
The vertices of triangle symbol Upper A prime Upper B prime Upper C prime△A′B′C′ are Upper A primeA′enter your response here​, Upper B primeB′enter your response here​, and Upper C primeC′enter your response here.
​(Simplify your answers. Type ordered​ pairs.)

Answer 1

To find the vertices of triangle \( A'B'C' \) after the translation \( T_{(-4,5)}( \triangle ABC) \), we will apply the translation to each vertex of triangle \( ABC \).

The translation \( T_{(-4,5)} \) means that we will subtract 4 from the x-coordinate and add 5 to the y-coordinate of each vertex.

Here are the vertices of triangle \( ABC \):

• \( A(2, -1) \)
• \( B(-2, -3) \)
• \( C(4, 3) \)

Now we will translate each vertex:

1. For Vertex A: \[ A' = (2 - 4, -1 + 5) = (-2, 4) \]

2. For Vertex B: \[ B' = (-2 - 4, -3 + 5) = (-6, 2) \]

3. For Vertex C: \[ C' = (4 - 4, 3 + 5) = (0, 8) \]

Now we can summarize the vertices of triangle \( A'B'C' \):

• \( A'(-2, 4) \)
• \( B'(-6, 2) \)
• \( C'(0, 8) \)

Therefore, the vertices of triangle \( A'B'C' \) are:

• \( A'(-2, 4) \)
• \( B'(-6, 2) \)
• \( C'(0, 8) \)

In conclusion:

• A' = (-2, 4)
• B' = (-6, 2)
• C' = (0, 8)