Question

20. Given A(4, 2) and B(-1, y) and the graph of line t below, find the value of y so that AB is perpendicular to t

Answer 1

Given two points, A(4,2) and B(-1,y), and a line with slope 4, the value of y that makes AB perpendicular to the line is 5/4.

​Two lines are perpendicular when their slopes multiply to -1. The slope of line t is 4, so the slope of line AB must be -1/4.

To find the slope of line AB, we use the formula:

m = (y2 - y1) / (x2 - x1)

Substituting in the coordinates of A and B, we get:

m = (y - 2) / (-1 - 4)

Setting this equal to -1/4, we get:

(y - 2) / -5 = -1/4

Multiplying both sides by -5, we get:

y - 2 = -1/4 * -5

Adding 2 to both sides, we get:

y = 5/4

The value of y that makes AB perpendicular to t is 5/4.

​

Answer 2

two lines are perpendicular when the multiplication of their slopes is equal to -1.

the slope of a line that passes through points (x1, y1) and (x2, y2) is computed as follows:

`m=\text{ }(y_2-y_1)/(x_2-x_1)`

Then, the slope of the lines that passes through A(4,2) and B(-1,y) is:

`m_1=(y-2)/(-1-4)=(y-2)/(-5)`

From the picture, t passes through (-1, 2) and (2,4), then its slope is:

`m_2=(4-2)/(2-(-1))=(2)/(3)`

Then

`\begin{gathered} m_1\cdot m_2=-1 \\ (y-2)/(-5)\cdot(2)/(3)=-1 \\ ((y-2)\cdot2)/(-15)=-1 \\ (y-2)\cdot2=(-1)\cdot(-15) \\ y-2=(15)/(2) \\ y=(15)/(2)+2 \\ y=(19)/(2) \end{gathered}`