Answer:
x = 4
Step-by-step explanation:
Label the points from A-F. I attached an image of how I labeled the points.
Notice that there are similar triangles here:
- [tex]\triangle ABD \sim \triangle FED[/tex]
- [tex]\triangle ACD \sim \triangle AEF[/tex]
Set up proportions to find the values of x, y, and z.
- [tex]\displaystyle \text{Equation I: }\frac{FD}{AD} = \frac{FE}{AB}[/tex]
- [tex]\displaystyle \text{Equation II: } \frac{A F}{AD}=\frac{EF}{CD}[/tex]
Let's make this problem simpler by incorporating only 2 variables instead of 3. Let's say that the total distance between the two poles is distance d. Instead of z, let's call it d - y.
Substitute known values or variables into the proportions.
- [tex]\displaystyle \text{Equation I: } \frac{d-y}{d} = \frac{x}{6}[/tex]
- [tex]\displaystyle \text{Equation II: }\frac{y}{d}= \frac{x}{12}[/tex]
Cross-multiply and simplify Equation I.
- [tex]6(d-y)=dx[/tex]
- [tex]6d-6y=dx[/tex]
Cross-multiply and simplify Equation II.
We now have two equations that are equal to dx, so we can set them equal to each other.
Add 6y to both sides of the equation.
Divide both sides of the equation by 6.
We can substitute this value of d back into either Equation I or II. I am going to substitute d into Equation II.
- [tex]\displaystyle \frac{y}{3y} = \frac{x}{12}[/tex]
Cross-multiply and simplify the equation.
Divide y from both sides of the equation.
Divide both sides of the equation by 3.
The guylines from the top of one pole to the bottom of the other cross at the height of 4 ft off the ground.
