The vertices of triangle symbol ABC△ABC are Aleft parenthesis 4 comma negative 2 right parenthesis(4,−2)​, Bleft parenthesis negative 1 comma negative 3 right parenthesis(−1,−3)​, and Cleft parenthesis 2 comma 1 right parenthesis(2,1). For the translation​ below, give the vertices of triangle symbol Upper A prime Upper B prime Upper C prime△A′B′C′.

Question

Upper T Subscript left angle negative 3 comma negative 5 right angle Baseline left parenthesis triangle symbol ABC right parenthesisT−3,−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 translate the vertices of triangle $ABC$ by the transformation $T(-3,-5)$, you need to subtract 3 from the x-coordinates and 5 from the y-coordinates of each vertex.

The original vertices of triangle $ABC$ are:

• $A(4,-2)$
• $B(-1,-3)$
• $C(2,1)$

Let's apply the translation to each vertex:

1. For vertex $A(4,-2)$: $$A^{\prime }=(4-3,-2-5)=(1,-7)$$

2. For vertex $B(-1,-3)$: $$B^{\prime }=(-1-3,-3-5)=(-4,-8)$$

3. For vertex $C(2,1)$: $$C^{\prime }=(2-3,1-5)=(-1,-4)$$

Now, let's summarize the vertices of triangle $A^{\prime }{B}^{\prime }{C}^{\prime }$:

• $A^{\prime }=(1,-7)$
• $B^{\prime }=(-4,-8)$
• $C^{\prime }=(-1,-4)$

Thus, the vertices of triangle $A^{\prime }{B}^{\prime }{C}^{\prime }$ are:

• $A^{\prime }(1,-7)$
• $B^{\prime }(-4,-8)$
• $C^{\prime }(-1,-4)$

In conclusion:

• $A^{\prime }=(1,-7)$
• $B^{\prime }=(-4,-8)$
• $C^{\prime }=(-1,-4)$