Question

monday (11/10/25)
write rules for the composition of translations.
translation 1:
translation 1:

Explanation:

Step1: Find coordinates of points

Let's assume \(A(-4,2)\), \(B(-1,1)\), \(C(-4,1)\) (from the left - hand triangle). After first translation (Translation 1), \(A'\) has coordinates \((1,3)\), \(B'\) has coordinates \((4,2)\), \(C'\) has coordinates \((1,2)\).
The rule for Translation 1: \((x,y)\to(x + 5,y+1)\) (because \(x\) - coordinate: \(-4+5 = 1\), \(y\) - coordinate: \(2 + 1=3\) for point \(A\)).

Step2: Find coordinates of final - translated points

After the second translation (Translation 2), \(A''\) has coordinates \((-4,-2)\), \(B''\) has coordinates \((-1,-3)\), \(C''\) has coordinates \((-4,-3)\).
If we consider the transformation from \(A'\) to \(A''\): \(x\) - coordinate: \(1-5=-4\), \(y\) - coordinate: \(3 - 5=-2\).
The rule for Translation 2: \((x,y)\to(x-5,y - 5)\)

Step3: Find the composition rule

If we first apply Translation 1 \((x,y)\to(x + 5,y+1)\) and then Translation 2 \((x,y)\to(x-5,y - 5)\).
Let \((x,y)\) be a general point. After Translation 1: \((x_1,y_1)=(x + 5,y+1)\). After Translation 2: \(x_2=x_1-5=(x + 5)-5=x\), \(y_2=y_1-5=(y + 1)-5=y-4\)

Answer:

The composition of the two translations is given by the rule \((x,y)\to(x,y - 4)\)