Answer:
[tex]x=-7[/tex]
Step-by-step explanation:
Remember that:
- Two lines are parallel if their slopes are the same.
- Two lines are perpendicular if their slopes are negative reciprocals.
- And two lines are neither (a.k.a intersecting) if they are neither parallel nor perpendicular.
We want to find the value of x such that [tex]\overline{BC}\parallel\overline{DE}[/tex].
Therefore, the slopes of BC and DE must be equivalent.
So, let's find the slope of BC first.
BC)
We can use the slope formula:
[tex]\displaystyle m=\frac{y_2-y_1}{x_2-x_1}[/tex]
Let B(17, 5) be (x₁, y₁) and let C(11, -3) be (x₂, y₂). Substitute:
[tex]\displaystyle m=\frac{-3-5}{11-17}[/tex]
Subtract:
[tex]m=-8/-6=4/3[/tex]
So, the slope of BC is 4/3.
DE)
Let D(-1, 2) be (x₁, y₁) and let E(x, -6) be (x, y₂). Substitute:
[tex]\displaystyle m=\frac{-6-2}{x-(-1)}[/tex]
We know that the two slopes must be equal. So, the slope of DE must also be 4/3. Substitute 4/3 for m:
[tex]\displaystyle \frac{4}{3}=\frac{-6-2}{x-(-1)}[/tex]
Solve for x. Simplify the right:
[tex]\displaystyle \frac{4}{3}=\frac{-8}{x+1}[/tex]
Cross multiply:
[tex]4(x+1)=-8(3)[/tex]
Multiply on the right:
[tex]4(x+1)=-24[/tex]
Divide both sides by 4:
[tex]x+1=-6[/tex]
Subtract 1 from both sides:
[tex]x=-7[/tex]
So, the value of x is -7.
And we're done!
