Line segment segment CD has a slope of negative start fraction two over three end fraction and contains point C(−3, −6). What is the y-coordinate of point Q(−1, y) if segment QC is perpendicular to line segment segment CD question mark

CD has slope 2/3

so, QC has slope -3/2

CD:
y+6 = 2/3 (x+3)

CQ (since it contains point C):
y+6 = -3/2 (x+3)
At x = -1,

y+6 = -3/2 (-1+3)
y+6 = -3
y = -9

Q is thus (-1,-9)

To find the y-coordinate of point Q, we can first find the equation of line segment CD using the slope-intercept form of a linear equation.

The slope-intercept form of a linear equation is given by y = mx + b, where m is the slope and b is the y-intercept.

Given that the slope of line segment CD is -2/3, we can substitute this value into the equation to get:

y = (-2/3)x + b

Next, we substitute the coordinates of point C(-3,-6) into the equation to find the value of b:

-6 = (-2/3)(-3) + b
-6 = 2 + b
b = -8

So, the equation of line segment CD is y = (-2/3)x - 8.

Since line segment QC is perpendicular to line segment CD, it means their slopes are negative reciprocals of each other. The negative reciprocal of -2/3 is 3/2.

Now, we can find the equation of line segment QC passing through point Q(-1, y) using the point-slope form of a linear equation.

The point-slope form of a linear equation is given by y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope.

Given that the slope of line segment QC is 3/2 and it passes through point Q(-1, y), we can substitute these values into the equation to get:

y - y1 = (3/2)(x - x1)

Substituting (-1, y) into the equation, we have:

y - y1 = (3/2)(x - (-1))
y - y1 = (3/2)(x + 1)
y - y1 = (3/2)x + 3/2

Simplifying further, we get:

y - y1 = (3/2)x + 3/2
y = (3/2)x + 3/2 + y1

Since we want to find the y-coordinate of point Q, we need to find the value of y when x = -1. Substituting x = -1 into the equation, we have:

y = (3/2)(-1) + 3/2 + y1
y = -3/2 + 3/2 + y1
y = 0 + y1
y = y1

Therefore, the y-coordinate of point Q is equal to the y-coordinate of point C, which is -6.