1. pia printed two maps of a walking trail. the length of the trail on the first map is 8 cm. the length of the trail on the second map is 6 cm. (a) 1 cm on the first map represents 2 km on the actual trail. what is the scale factor from the map to the actual trail? what is the length of the actual trail? (b) a landmark on the first map is a triangle with side lengths of 3 mm, 4 mm, and 5 mm. what is the scale factor from the first map to the second map? what are the side lengths of the landmark on the second map? show your work.

We know that

Pia printed two trail maps:
Trail on the first map = 8 cm
Trail on the second map = 6 cm
Scale on first map = 1 cm : 2 km

A) What is the scale factor from the map to the actual trail? 
For the first map, the scale factor is 1 cm: 2km.
Therefore
the actual trail is 8 centimeters * 2 kilometers = 16 km. 
The scale factor of the second map is 16 km / 6 cm = 2.67 km / cm 
the scale on the 2nd map is 1 cm represent 2.67 km in the actual trail

B) The length of the actual trail is 16 kilometers. 

C) What is the scale factor from the first map to the second map?
=2/2.67> 0.75 km/km
the scale factor from the first map to the second map is 0.75 km in the first map is 1 km in the second map

D) What are the side lengths of the landmark on the second map?
we know that
in the first map the side lengths of the landmark are 3 mm, 4 mm, and 5 mm

step 1
convert mm in the map1 to Km in the actual
the scale factor is 1 cm: 2km

for 3 mm> 0.3 cm
1 cm> 2 km
0.3 cm> X
X=0.3*2> 0.6 km

for 4 mm> 0.4 cm
1 cm> 2 km
0.4 cm> X
X=0.4*2> 0.8 km

for 5 mm> 0.5 cm
1 cm> 2 km
0.5 cm> X
X=0.5*2> 1 km

step 2
convert Km in the actual to cm in the map 2
the scale on the 2nd map is 1 cm represent 2.67 km in the actual trail

for 0.6 km
1 cm> 2.67 km
X> 0.6 Km
X=0.6/2.67> x=0.2247 cm> 2.25 mm

for 0.8 km
1 cm> 2.67 km
X> 0.8 Km
X=0.8/2.67> x=0.2996 cm> 3 mm

for 1 km
1 cm> 2.67 km
X> 1 Km
X=1/2.67> x=0.3745 cm> 3.75 mm

the answer Part D) is
the side lengths of the landmark on the second map are
2.25 mm 3 mm and 3.75 mm