Question

dilating a segment in the coordinate plane
the rule ( d_{o,0.25}(x,y)\to(0.25x,0.25y) ) is applied to the segment lm to make an image of segment ( lm ), not shown.
the coordinates of ( l ) in the image are ( (-1,2) ).
the coordinates of ( m ) in the image are ( (1,2) ).
the length, ( lm ), is 4.
the slope of the original segment and dilated segment are both zero

Explanation:

Step1: Find coordinates of \(L'\) and \(M'\)

Given \(L(- 4,8)\) and \(M(4,8)\), using the dilation rule \(D_{O,0.25}(x,y)=(0.25x,0.25y)\)
For \(L\): \(x=-4,y = 8\), then \(L'=(0.25\times(-4),0.25\times8)=(-1,2)\)
For \(M\): \(x = 4,y=8\), then \(M'=(0.25\times4,0.25\times8)=(1,2)\)

Step2: Calculate the length of \(L'M'\)

The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). Here \(x_1=-1,y_1 = 2,x_2=1,y_2 = 2\)
\(d=\sqrt{(1-(-1))^2+(2 - 2)^2}=\sqrt{(2)^2+0^2}=2\)

Answer:

The length of \(L'M'\) is \(2\)