Question

do the math

1. identify the coordinates of △abc under the translation that moves the triangle 3 units up and 2 units to the right.

a=(-1 + ,3 + 3)
=(,)
b=(+2,-1 + )=(,)
c=(3 + ,+3)=(,)
identify the coordinates of △abc under each translation.

1. 5 units up
2. 2 units to the right
3. 1 unit down
4. 6 units to the left
5. 3 units to the right and 6 units up
6. 10 units to the left and 9 units up
7. 6 units to the left, 4 units up, and 2 units to the right
8. a floor pattern begins with a black square tile that has vertices at (0, 0), (5, 0), (0, 5), and (5, 5). the next black tile is located 5 units to the right and 5 units up. what are the vertices of the next tile?
9. combinations of translations such as “2 units to the right, 3 units down, 4 units to the left, and then 6 units up” can be complicated. explain why all combinations of translations can be reduced to at most two movements, one horizontal and the other vertical.

remember
a translation right adds to the x - coordinate.
a translation left subtracts from the x - coordinate.
a translation up adds to the y - coordinate.
a translation down subtracts from the y - coordinate.

Explanation:

Step1: Identify original coordinates

Assume \(A(-1,3)\), \(B(0, - 1)\), \(C(3,0)\)

Step2: Calculate \(A'\) for first - given translation

A translation 3 units up and 2 units to the right:
For \(A(-1,3)\), \(A'=(-1 + 2,3 + 3)=(1,6)\)

Step3: Calculate \(B'\) for first - given translation

For \(B(0,-1)\), \(B'=(0 + 2,-1 + 3)=(2,2)\)

Step4: Calculate \(C'\) for first - given translation

For \(C(3,0)\), \(C'=(3+2,0 + 3)=(5,3)\)

1. For a translation 5 units up:

If \(A(-1,3)\), \(A_1=(-1,3 + 5)=(-1,8)\)
If \(B(0,-1)\), \(B_1=(0,-1 + 5)=(0,4)\)
If \(C(3,0)\), \(C_1=(3,0 + 5)=(3,5)\)

1. For a translation 2 units to the right:

If \(A(-1,3)\), \(A_2=(-1+2,3)=(1,3)\)
If \(B(0,-1)\), \(B_2=(0 + 2,-1)=(2,-1)\)
If \(C(3,0)\), \(C_2=(3+2,0)=(5,0)\)

1. For a translation 1 unit down:

If \(A(-1,3)\), \(A_3=(-1,3-1)=(-1,2)\)
If \(B(0,-1)\), \(B_3=(0,-1 - 1)=(0,-2)\)
If \(C(3,0)\), \(C_3=(3,0-1)=(3,-1)\)

1. For a translation 6 units to the left:

If \(A(-1,3)\), \(A_4=(-1-6,3)=(-7,3)\)
If \(B(0,-1)\), \(B_4=(0-6,-1)=(-6,-1)\)
If \(C(3,0)\), \(C_4=(3-6,0)=(-3,0)\)

1. For a translation 3 units to the right and 6 units up:

If \(A(-1,3)\), \(A_5=(-1 + 3,3+6)=(2,9)\)
If \(B(0,-1)\), \(B_5=(0 + 3,-1 + 6)=(3,5)\)
If \(C(3,0)\), \(C_5=(3+3,0 + 6)=(6,6)\)

1. For a translation 10 units to the left and 9 units up:

If \(A(-1,3)\), \(A_6=(-1-10,3 + 9)=(-11,12)\)
If \(B(0,-1)\), \(B_6=(0-10,-1 + 9)=(-10,8)\)
If \(C(3,0)\), \(C_6=(3-10,0 + 9)=(-7,9)\)

1. For a translation 6 units to the left, 4 units up, and 2 units to the right:

Net left - right movement: \(-6 + 2=-4\) (4 units to the left)
If \(A(-1,3)\), \(A_7=(-1-4,3 + 4)=(-5,7)\)
If \(B(0,-1)\), \(B_7=(0-4,-1 + 4)=(-4,3)\)
If \(C(3,0)\), \(C_7=(3-4,0 + 4)=(-1,4)\)

1. For the square with vertices \((0,0)\), \((5,0)\), \((0,5)\), \((5,5)\) and a translation 5 units to the right and 5 units up:

The new vertices are \((0 + 5,0 + 5)=(5,5)\), \((5+5,0 + 5)=(10,5)\), \((0 + 5,5 + 5)=(5,10)\), \((5+5,5 + 5)=(10,10)\)

1. All combinations of translations can be reduced to at most two movements (one horizontal and one vertical) because the horizontal movements (left - right) can be combined into a single net horizontal movement (by adding the displacements, where right - movement is positive and left - movement is negative), and the vertical movements (up - down) can be combined into a single net vertical movement (by adding the displacements, where up - movement is positive and down - movement is negative).

Answer:

1. \(A'=(1,6)\), \(B'=(2,2)\), \(C'=(5,3)\)
2. \(A=(-1,8)\), \(B=(0,4)\), \(C=(3,5)\)
3. \(A=(1,3)\), \(B=(2,-1)\), \(C=(5,0)\)
4. \(A=(-1,2)\), \(B=(0,-2)\), \(C=(3,-1)\)
5. \(A=(-7,3)\), \(B=(-6,-1)\), \(C=(-3,0)\)
6. \(A=(2,9)\), \(B=(3,5)\), \(C=(6,6)\)
7. \(A=(-11,12)\), \(B=(-10,8)\), \(C=(-7,9)\)
8. \(A=(-5,7)\), \(B=(-4,3)\), \(C=(-1,4)\)
9. \((5,5)\), \((10,5)\), \((5,10)\), \((10,10)\)
10. All horizontal displacements can be combined into one net horizontal displacement and all vertical displacements can be combined into one net vertical displacement.