Question

S(4;8) and T(-2;6) are joined by a straight line. Determine the equation of a line that is perpendicular to ST and goes through the midpoint between S and T

Answer 1

the equation of the line that is perpendicular to ST and goes through the midpoint between S and T is 3x + y = 10

Explanation:

To find the equation of a line that is perpendicular to ST and goes through the midpoint between S and T, we can follow these steps:1.

Find the slope of the line ST:- The slope of a line can be found using the formula: slope = (change in y-coordinates) / (change in x-coordinates).- Given S(4,8) and T(-2,6), the change in y-coordinates is 6 - 8 = -2 and the change in x-coordinates is -2 - 4 = -6.- Therefore, the slope of line ST is -2 / -6 = 1/3.2.

Determine the negative reciprocal of the slope of ST to find the slope of the line perpendicular to ST:- The negative reciprocal of a slope is obtained by flipping the fraction and changing its sign.- So, the negative reciprocal of 1/3 is -3/1 or simply -3.3.

Find the midpoint between S and T:- The midpoint of two points can be found using the midpoint formula: ((x₁ + x₂)/2, (y₁ + y₂)/2).- Using S(4,8) and T(-2,6), we have ((4 + -2)/2, (8 + 6)/2) = (1, 7).4.

Use the slope and the midpoint to determine the equation of the line:- We can use the point-slope form of a line, which is y - y₁ = m(x - x₁), where (x₁, y₁) is the midpoint and m is the slope.- Substituting the values, we get y - 7 = -3(x - 1).- Expanding and rearranging the equation, we have y - 7 = -3x + 3.- Finally, we can rewrite the equation in the standard form as 3x + y = 10.Therefore, the equation of the line that is perpendicular to ST and goes through the midpoint between S and T is 3x + y = 10.

Answer 2

`y=-3x+10`

Explanation:

To determine the equation of the line that is perpendicular to ST and goes through the midpoint between points S and T, we need to find the slope of ST and the midpoint of ST.

To find the slope of ST, substitute the points S(4, 8) and T(-2, 6) into the slope formula.

`\boxed{\begin{minipage}{8cm}\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}}`

`\textsf{Slope}\:(m)=(y_T-y_S)/(x_T-x_S)=(6-8)/(-2-4)=(-2)/(-6)=(1)/(3)`

If two lines are perpendicular to each other, their slopes are negative reciprocals. Therefore, the slope of the line that is perpendicular to ST is m = -3.

To find the midpoint between S and T, substitute the points S(4, 8) and T(-2, 6) into the midpoint formula.

`\boxed{\begin{array}{l}\underline{\sf Midpoint \;formula}\\\\M(x,y) =\left((x_2+x_1)/(2),(y_2+y_1)/(2)\right)\\\\\textsf{where $(x_1,y_1)$ and $(x_2,y_2)$ are the endpoints.}\\\end{array}}`

Therefore:

`M(x,y)=\left((x_T+x_S)/(2),(y_T+y_S)/(2)\right)`

`M(x,y)=\left((-2+4)/(2),(6+8)/(2)\right)`

`M(x,y)=\left((2)/(2),(14)/(2)\right)`

`M(x,y)=\left(1,7\right)`

To determine the equation of a line that is perpendicular to ST and goes through the midpoint between S and T, substitute the found slope m = -3 and the midpoint (1, 7) into the point-slope formula:

`\begin{aligned}y-y_1&=m(x-x_1)\\y-7&=-3(x-1)\\y-7&=-3x+3\\y&=-3x+10\end{aligned}`

Therefore, the equation of the line is:

`\large\boxed{\boxed{y=-3x+10}}`