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Triangle [tex]$PQR$[/tex] has coordinates [tex]$P(-8,3)$[/tex], [tex]$Q(-8,6)$[/tex], and [tex]$R(-3,6)$[/tex]. If the triangle is translated by using the rule [tex]$(x, y) \rightarrow (x+4, y-6)$[/tex], what are the coordinates of triangle [tex]$P^{\prime}Q^{\prime}R^{\prime}$[/tex]?

A. [tex]$P^{\prime}(-12,9)$[/tex], [tex]$Q^{\prime}(-12,12)$[/tex], [tex]$R^{\prime}(-7,12)$[/tex]
B. [tex]$P^{\prime}(-4,7)$[/tex], [tex]$Q^{\prime}(-14,0)$[/tex], [tex]$R^{\prime}(1,10)$[/tex]
C. [tex]$P^{\prime}(-4,9)$[/tex], [tex]$Q^{\prime}(-4,12)$[/tex], [tex]$R^{\prime}(1,12)$[/tex]
D. [tex]$P^{\prime}(-4,-3)$[/tex], [tex]$Q^{\prime}(-4,0)$[/tex], [tex]$R^{\prime}(1,0)$[/tex]



Answer :

GinnyAnswer
To determine the coordinates of the translated triangle [tex]\( P'Q'R' \)[/tex] from the given translation rule [tex]\((x, y) \rightarrow (x+4, y-6)\)[/tex], follow these step-by-step instructions:

1. Starting Coordinates:
- The original coordinates of the triangle [tex]\(PQR\)[/tex] are:
[tex]\[ P(-8, 3), \quad Q(-8, 6), \quad R(-3, 6) \][/tex]

2. Translation Rule:
- According to the translation rule [tex]\((x, y) \rightarrow (x+4, y-6)\)[/tex]:

3. Translating Each Point:
- For point [tex]\(P(-8, 3)\)[/tex]:
[tex]\[ x + 4 = -8 + 4 = -4 \][/tex]
[tex]\[ y - 6 = 3 - 6 = -3 \][/tex]
Therefore, the coordinates of [tex]\(P'\)[/tex] are:
[tex]\[ P'(-4, -3) \][/tex]

- For point [tex]\(Q(-8, 6)\)[/tex]:
[tex]\[ x + 4 = -8 + 4 = -4 \][/tex]
[tex]\[ y - 6 = 6 - 6 = 0 \][/tex]
Therefore, the coordinates of [tex]\(Q'\)[/tex] are:
[tex]\[ Q'(-4, 0) \][/tex]

- For point [tex]\(R(-3, 6)\)[/tex]:
[tex]\[ x + 4 = -3 + 4 = 1 \][/tex]
[tex]\[ y - 6 = 6 - 6 = 0 \][/tex]
Therefore, the coordinates of [tex]\(R'\)[/tex] are:
[tex]\[ R'(1, 0) \][/tex]

4. Conclusion:
- The translated coordinates of triangle [tex]\(P'Q'R'\)[/tex] are:
[tex]\[ P'(-4, -3), \quad Q'(-4, 0), \quad R'(1, 0) \][/tex]

From the choices provided, the coordinates [tex]\((-4, -3), (-4, 0), (1, 0)\)[/tex] correspond to the triangle [tex]\(P'Q'R'\)[/tex]. Therefore, the correct answer is:
[tex]\[ \boxed{P'(-4, -3), Q'(-4, 0), R'(1, 0)} \][/tex]

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