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The diagram shows two campsites on opposite ends of a lake and two triangles formed by intersecting segments from the campsites.
Campsites F and G are on opposite sides of a lake. A survey crew made the measurements shown on the diagram. What is the distance between the two campsites? The diagram is not drawn to scale. The triangles are proportional.
To find the distance between campsites F and G, we can use the concept of similar triangles and scale factors.
Let's first examine triangle FHO and triangle GKP. These two triangles are similar since they share angle HOK and angle FHO is congruent to angle GKP. This means that the corresponding sides are proportional.
Using the measurements given in the diagram:
FH = 40 ft
KP = 45 ft
OH = 60 ft
OK = 90 ft
HP = 72 ft
Let x be the distance between campsites F and G. We can set up a proportion using the corresponding sides of the triangles:
FH / KP = OH / OK
40 / 45 = 60 / 90
4/5 = 2/3
Now, let's find the length of FG using the proportion of HP to KP (which is also equal to FG / KP):
HP / KP = FG / KP
72 / 45 = x / 45
72 / 45 = x / 45
8 / 5 = x / 45
x = (8/5) * 45
x = 72 ft
Therefore, the distance between campsites F and G is 72 feet.