Question

dilating a segment in the coordinate plane
the rule ( d_{0,0.25}(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 ( (-16,32) ).
the coordinates of ( m ) in the image are ( (4,8) ).
the length, ( lm ), is
the slope of the original segmentand dilated segment
are

Explanation:

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

From the graph, \(L(- 16,8)\) and \(M(4,8)\)

Step2: Calculate length of \(LM\)

Use the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). Here \(x_1=-16,y_1 = 8,x_2 = 4,y_2 = 8\). So \(LM=\sqrt{(4-(-16))^2+(8 - 8)^2}=\sqrt{(20)^2+0}=20\)

Step3: Calculate length of \(L'M'\)

Since the dilation rule is \(D_{O,0.25}(x,y)=(0.25x,0.25y)\), the scale factor \(k = 0.25\). The length of a dilated segment \(L'M'=k\times LM\). So \(L'M'=0.25\times20 = 5\)

Step4: Calculate slope of \(LM\)

Use the slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\). For \(L(-16,8)\) and \(M(4,8)\), \(m_{LM}=\frac{8 - 8}{4-(-16)}=0\)

Step5: Calculate slope of \(L'M'\)

\(L'(-4,2)\) and \(M'(1,2)\). Using the slope formula \(m_{L'M'}=\frac{2 - 2}{1-(-4)}=0\)

Answer:

The length of \(L'M'\) is \(5\). The slope of the original segment and dilated segment is \(0\)