Question

translations
what is the rule for the translation shown?
1.
2.
the vertices of △abc are a(2, -3), b(-3, -5), and c(4, 1). for each translation, give the vertices of △abc.

1. t_{(-2,5)}(△abc)
2. t_{(-4,-1)}(△abc)
3. t_{(4,6)}(△abc)

Explanation:

Step1: Recall translation rule

A translation $T_{(a,b)}$ moves a point $(x,y)$ to $(x + a,y + b)$.

Step2: Find vertices for $T_{(-2,5)}$

For point $A(2,-3)$:
$x=2,y = - 3,a=-2,b = 5$. New - coordinates are $(2+( - 2),-3 + 5)=(0,2)$.
For point $B(-3,-5)$:
$x=-3,y=-5,a=-2,b = 5$. New - coordinates are $(-3+( - 2),-5 + 5)=(-5,0)$.
For point $C(4,1)$:
$x = 4,y = 1,a=-2,b = 5$. New - coordinates are $(4+( - 2),1 + 5)=(2,6)$.
So, $A'(0,2),B'(-5,0),C'(2,6)$.

Step3: Find vertices for $T_{(-4,-1)}$

For point $A(2,-3)$:
$x = 2,y=-3,a=-4,b=-1$. New - coordinates are $(2+( - 4),-3+( - 1))=(-2,-4)$.
For point $B(-3,-5)$:
$x=-3,y=-5,a=-4,b=-1$. New - coordinates are $(-3+( - 4),-5+( - 1))=(-7,-6)$.
For point $C(4,1)$:
$x = 4,y = 1,a=-4,b=-1$. New - coordinates are $(4+( - 4),1+( - 1))=(0,0)$.
So, $A'(-2,-4),B'(-7,-6),C'(0,0)$.

Step4: Find vertices for $T_{(4,6)}$

For point $A(2,-3)$:
$x = 2,y=-3,a = 4,b = 6$. New - coordinates are $(2 + 4,-3+6)=(6,3)$.
For point $B(-3,-5)$:
$x=-3,y=-5,a = 4,b = 6$. New - coordinates are $(-3 + 4,-5 + 6)=(1,1)$.
For point $C(4,1)$:
$x = 4,y = 1,a = 4,b = 6$. New - coordinates are $(4 + 4,1 + 6)=(8,7)$.
So, $A'(6,3),B'(1,1),C'(8,7)$.

Answer:

1. $A'(0,2),B'(-5,0),C'(2,6)$
2. $A'(-2,-4),B'(-7,-6),C'(0,0)$
3. $A'(6,3),B'(1,1),C'(8,7)$