Question

1. you are visiting washington, d.c. from the american history museum, you walk 5 blocks east and 1 block south to the air and space museum. then you walk 8 blocks west to the washington monument. write a translation rule to describe the total distance you moved. (let north = up, south = down, east = right, and west = left)

Explanation:

Step1: Determine the translation rule for east - west and north - south movements

In a coordinate system, if we assume the starting point as \((x,y)\). Moving east \(a\) units and west \(b\) units affects the \(x\) - coordinate. Moving north \(c\) units and south \(d\) units affects the \(y\) - coordinate. The general translation rule is \((x,y)\to(x + a - b,y + c - d)\). Here, \(a = 5\) (east), \(b=8\) (west), \(c = 0\) (no north movement), \(d = 1\) (south).

Step2: Calculate the translation rule

For the \(x\) - coordinate: \(x\to x+(5 - 8)=x - 3\). For the \(y\) - coordinate: \(y\to y+(0 - 1)=y - 1\). So the translation rule is \((x,y)\to(x - 3,y - 1)\)

Step3: Calculate the total distance using the distance formula

The distance formula between two points \((x_1,y_1)\) (starting point, assume \((x_1,y_1)=(0,0)\)) and \((x_2,y_2)\) (ending point \((x_2=x - 3,y_2=y - 1)\)) is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). Substituting \(x_1 = 0,y_1 = 0,x_2=-3,y_2=-1\), we get \(d=\sqrt{(- 3-0)^2+(-1 - 0)^2}=\sqrt{9 + 1}=\sqrt{10}\)

Answer:

The translation rule is \((x,y)\to(x - 3,y - 1)\) and the total distance moved is \(\sqrt{10}\) blocks.