Answer: 4.6 mi
Step-by-step explanation:
Suppose the points A, B, C and S shows Alba, Blare, Cray and Service station respectively,
Then, We have to find out the line segment CS = x = ?
Now, In the triangles ACB and CSB,
[tex]\angle ACB\cong \angle CSB[/tex] ( Right angles )
[tex]\angle ABC\cong \angle CBS[/tex] ( Reflexive angles )
Thus, by AA similarity postulate,
[tex]\triangle ACB\sim\triangle CSB[/tex]
By the property of similar triangles,
[tex]\frac{AB}{CB}=\frac{AC}{CS}[/tex]
[tex]\implies \frac{13}{5}=\frac{12}{x}[/tex]
[tex]\implies 13x=5\times 12 [/tex]
[tex]\implies 13x = 60[/tex]
[tex]\implies x =\frac{60}{13}=4.61539\approx 4.6[/tex]
Hence, the approximate length of the new road = 4.6 miles
