Question

In Triangle JKL. J(2,5), K(6,-7),L(-6,1) write the equation of the perpendicular bisector of LK

Answer 1

`y=(3)/(2)x-3`

Explanation:

Given vertices of triangle JKL:

• J = (2, 5)
• K = (6, -7)
• L = (-6, 1)

A perpendicular bisector is a line that intersects another line segment at 90°, dividing it into two equal parts.

To find the equation of the perpendicular bisector of LK, find the midpoint between the two points and the slope of the line LK.

`\boxed{\begin{minipage}{7.4 cm}\underline{Midpoint between two points}\\\\Midpoint $=\left((x_2+x_1)/(2),(y_2+y_1)/(2)\right)$\\\\\\where $(x_1,y_1)$ and $(x_2,y_2)$ are the endpoints.\\\end{minipage}}`

Find the midpoint of LK:

`\implies \textsf{Midpoint}=\left((x_K+x_L)/(2),(y_K+y_L)/(2)\right)`

`\implies \textsf{Midpoint}=\left((6+(-6))/(2),(-7+1)/(2)\right)`

`\implies \textsf{Midpoint}=\left((0)/(2),(-6)/(2)\right)`

`\implies \textsf{Midpoint}=\left(0,-3\right)`

`\boxed{\begin{minipage}{4.4cm}\underline{Slope Formula}\\\\Slope $(m)=(y_2-y_1)/(x_2-x_1)$\\\\where $(x_1,y_1)$ and $(x_2,y_2)$ \\are two points on the line.\\\end{minipage}}`

Find the slope of the line LK:

`\implies \textsf{Slope}\;(m)=(y_K-y_L)/(x_K-x_L)`

`\implies \textsf{Slope}\;(m)=(-7-1)/(6-(-6))`

`\implies \textsf{Slope}\;(m)=(-8)/(12)`

`\implies \textsf{Slope}\;(m)=-(2)/(3)`

If two lines are perpendicular to each other, their slopes are negative reciprocals. Therefore, the slope of the perpendicular bisector is ³/₂.

`\boxed{\begin{minipage}{5.8 cm}\underline{Point-slope form of a linear equation}\\\\$y-y_1=m(x-x_1)$\\\\where:\\ \phantom{ww}$\bullet$ $m$ is the slope. \\ \phantom{ww}$\bullet$ $(x_1,y_1)$ is a point on the line.\\\end{minipage}}`

Substitute the found midpoint and perpendicular slope into the point-slope formula to write the equation of the perpendicular bisector of LK:

`\implies y-(-3)=(3)/(2)(x-0)`

`\implies y+3=(3)/(2)x`

`\implies y=(3)/(2)x-3`

Answer 2

• The perpendicular bisector of LK is y = 1.5x - 3

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Find the slope of LK:

• m = (- 7 - 1) / (6 - (-6)) = - 8/12 = - 2/3

Find the midpoint of LK:

• x - coordinate: x = (6 - 6)/2 = 0,
• y - coordinate: y = (- 7 + 1)/2 = - 3.

Perpendicular slope is:

• -1/m = - 1/( - 2/3) = 3/2 = 1.5

The line is:

• y - (-3) = 1.5(x - 0)
• y + 3 = 1.5x
• y = 1.5x - 3