Question
The composition transformation that maps $ΔABC$ to $ΔA″B″C″$ is a translation defined by the rule $(x, y) → (x+9, y−9)$. This single translation combines the two successive translations applied to the triangle.
Answers
GPT-4o mini
To find the overall translation that combines two successive translations, we first need to understand how each translation affects the coordinates of points in the triangle ΔABC.
Let's denote the first translation as \( T_1 \) and the second translation as \( T_2 \).
1. **First Translation \( T_1 \)**:
This translation is defined by a rule that maps each point \( (x, y) \) to \( (x + a, y + b) \). We can represent the first translation generally as:
\[
T_1: (x, y) \to (x + a_1, y + b_1)
\]
where \( a_1 \) and \( b_1 \) are the constants of the first translation.
2. **Second Translation \( T_2 \)**:
Similarly, the second translation is defined by the rule:
\[
T_2: (x, y) \to (x + a_2, y + b_2)
\]
where \( a_2 \) and \( b_2 \) are the constants of the second translation.
To find the overall effect of these two translations, we need to apply the first translation \( T_1 \) followed by the second translation \( T_2 \). This means we first apply \( T_1 \) to a point \( (x, y) \), and then take the result and apply \( T_2 \) to it.
If we start from a point \( (x, y) \):
1. After applying \( T_1 \), we get:
\[
T_1(x, y) = (x + a_1, y + b_1)
\]
2. Then, we apply \( T_2 \) to this result:
\[
T_2(T_1(x, y)) = T_2(x + a_1, y + b_1) = ((x + a_1) + a_2, (y + b_1) + b_2)
\]
This simplifies to:
\[
(x + (a_1 + a_2), y + (b_1 + b_2))
\]
Thus, the overall translation that combines both \( T_1 \) and \( T_2 \) is given by the rule:
\[
(x, y) \to (x + (a_1 + a_2), y + (b_1 + b_2))
\]
In your specific case, you mentioned that the overall translation is given by the rule \( (x, y) → (x + 9, y - 9) \). This indicates:
- The total change in the \( x \)-coordinate is \( 9 \), so \( a_1 + a_2 = 9 \).
- The total change in the \( y \)-coordinate is \( -9 \), so \( b_1 + b_2 = -9 \).
To summarize, the overall composition transformation that combines the two successive translations into a single translation can be expressed as:
\[
(x, y) \to (x + 9, y - 9)
\]
where the net effect is a translation to the right by 9 units and downward by 9 units.
Let's denote the first translation as \( T_1 \) and the second translation as \( T_2 \).
1. **First Translation \( T_1 \)**:
This translation is defined by a rule that maps each point \( (x, y) \) to \( (x + a, y + b) \). We can represent the first translation generally as:
\[
T_1: (x, y) \to (x + a_1, y + b_1)
\]
where \( a_1 \) and \( b_1 \) are the constants of the first translation.
2. **Second Translation \( T_2 \)**:
Similarly, the second translation is defined by the rule:
\[
T_2: (x, y) \to (x + a_2, y + b_2)
\]
where \( a_2 \) and \( b_2 \) are the constants of the second translation.
To find the overall effect of these two translations, we need to apply the first translation \( T_1 \) followed by the second translation \( T_2 \). This means we first apply \( T_1 \) to a point \( (x, y) \), and then take the result and apply \( T_2 \) to it.
If we start from a point \( (x, y) \):
1. After applying \( T_1 \), we get:
\[
T_1(x, y) = (x + a_1, y + b_1)
\]
2. Then, we apply \( T_2 \) to this result:
\[
T_2(T_1(x, y)) = T_2(x + a_1, y + b_1) = ((x + a_1) + a_2, (y + b_1) + b_2)
\]
This simplifies to:
\[
(x + (a_1 + a_2), y + (b_1 + b_2))
\]
Thus, the overall translation that combines both \( T_1 \) and \( T_2 \) is given by the rule:
\[
(x, y) \to (x + (a_1 + a_2), y + (b_1 + b_2))
\]
In your specific case, you mentioned that the overall translation is given by the rule \( (x, y) → (x + 9, y - 9) \). This indicates:
- The total change in the \( x \)-coordinate is \( 9 \), so \( a_1 + a_2 = 9 \).
- The total change in the \( y \)-coordinate is \( -9 \), so \( b_1 + b_2 = -9 \).
To summarize, the overall composition transformation that combines the two successive translations into a single translation can be expressed as:
\[
(x, y) \to (x + 9, y - 9)
\]
where the net effect is a translation to the right by 9 units and downward by 9 units.