Question

the endpoints of segment mn have coordinates (0, 0) and (5, 1). the endpoints of segment ab have coordinates (1 1 2 , 2 1 4 ) and (−2 1 4 , k ). for what value of k are these segments parallel?

Answer 1

`k=(3)/(2)`

Explanation:

`\boxed{\begin{minipage}{4.3 cm}\underline{Slope formula}\\\\$m=(y_2-y_1)/(x_2-x_1)$\\\\where $(x_1,y_1)$ and $(x_2,y_2)$ \\ are points on the line.\end{minipage}}`

Segment MN

• M = (0, 0)
• N = (5, 1)

`\implies \textsf{slope}=(y_N-y_M)/(x_N-x_M)=(1-0)/(5-0)=(1)/(5)`

Segment AB

• A = (1 ¹/₂, 2 ¹/₄)
• B = (-2 ¹/₄, k)

`\implies \textsf{slope}=(y_B-y_A)/(x_B-x_A)=(k-2 (1)/(4))/(-2(1)/(4)-1(1)/(2))=(k-2 (1)/(4))/(-2(1)/(4)-1(2)/(4))=(k-2 (1)/(4))/(-3(3)/(4))`

Parallel lines have the same slope.

Therefore, equate the found slopes for the two segments and solve for k:

`\implies (k-2 (1)/(4))/(-3(3)/(4))=(1)/(5)`

Cross multiply:

`\implies 5 \left(k-2 (1)/(4)\right)=-3(3)/(4)`

`\implies 5k-10 (5)/(4)=-3(3)/(4)`

`\implies 5k-11 (1)/(4)=-3(3)/(4)`

Add 11 ¹/₄ to both sides:

`\implies 5k-11 (1)/(4)+11 (1)/(4)=-3(3)/(4)+11 (1)/(4)`

`\implies 5k=(-3+11)+\left(-(3)/(4)+(1)/(4)\right)`

`\implies 5k=8-(1)/(2)`

`\implies 5k=7(1)/(2)`

Divide both sides by 5:

`\implies 5k / 5=7(1)/(2) / 5`

`\implies 5k / 5=(15)/(2) / 5`

`\implies k=(3)/(2)`

Therefore, the value of k for which the two segments are parallel is ³/₂.

Answer 2

• When k = 1.5 the given segments are parallel

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Parallel lines have equal slopes.

Find the slope of the first line, mn:

• 
  `m = \cfrac{y_2-y_1}{x_2-x_1}= \cfrac{1-0}{5-0}=\cfrac{1}{5}`

Find the slope of the second line, ab:

• 
  `m= \cfrac{y_2-y_1}{x_2-x_1}= \cfrac{k-2.25}{-2.25-1.5}=\cfrac{k-2.25}{-3.75}`

Compare the slopes and find the value of k:

• 
  `\cfrac{k-2.25}{-3.75}=\cfrac{1}{5}`
• 
  `k-2.25=-\cfrac{3.75}{5}`
• 
  `k-2.25=-0.75`
• 
  `k=2.25-0.75`
• 
  `k=1.5`