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Sagot :
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.
- [tex]12y=dx[/tex]
We now have two equations that are equal to dx, so we can set them equal to each other.
- [tex]6d-6y=12y[/tex]
Add 6y to both sides of the equation.
- [tex]6d=18y[/tex]
Divide both sides of the equation by 6.
- [tex]d=3y[/tex]
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.
- [tex]12y=3xy[/tex]
Divide y from both sides of the equation.
- [tex]12=3x[/tex]
Divide both sides of the equation by 3.
- [tex]4=x[/tex]
The guylines from the top of one pole to the bottom of the other cross at the height of 4 ft off the ground.
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