Problem Solving

BTGmoderatorLU wrote:The line represented by the equation y = 4 - 2x is the perpendicular bisector of the line segment RP. If R has the coordinates (4, 1), what are the coordinates of point P?

A. (-4, 1)
B. (-2, 2)
C. (0, 1)
D. (0, -1)
E. (2, 0)

The OA is D.

Please, can anyone assist me with this PS question? I don't know how can I solve it. I need help. Thanks!

The equation provided by the question stem, y = 4 - 2x, is written in slope-intercept form, y = mx + b.
In the slope-intercept equation of a line, the coefficient of the x-term represents the line's slope, and 'b' represents the line's y-intercept.
Thus, the slope of the line is -2, and the line intersects the y-axis at point (0, 4).
Use this information to quickly sketch the line in an xy-coordinate plane, so that it passes through points (0, 4), (1, 2), (2, 0), and so on.

Line segment RP is perpendicular to the line with equation 'y = 4 - 2x'.
The slopes of perpendicular lines are negative reciprocals, so the slope of line segment RP must be 1/2.
Plot point R at coordinates (4, 1), and then use the slope of line segment RP to quickly sketch a line passing through points (4, 1), (2, 0), and (0, -1).

The line and line segment intersect at point (2, 0).
Since the line bisects line segment RP, the distance from the line to point R must be equal to the distance from the line to point P.
Point R is 2 units to the right and one unit above point (2, 0), so point P must be 2 units to the left and one unit below point (2, 0).

The correct answer is choice D.

BTGmoderatorLU wrote:The line represented by the equation y = 4 - 2x is the perpendicular bisector of the line segment RP. If R has the coordinates (4, 1), what are the coordinates of point P?

A. (-4, 1)
B. (-2, 2)
C. (0, 1)
D. (0, -1)
E. (2, 0)

The OA is D.

Please, can anyone assist me with this PS question? I don't know how can I solve it. I need help. Thanks!

We see that line segment RP will have a slope of 1/2 since it's perpendicular to the line y = 4 - 2x. We can examine the answer choices and see which coordinates will make a slope of 1/2 with (4, 1). Also, keep in mind that the perpendicular bisector of a line segment will also pass through the midpoint of the line segment.

A) (-4, 1)

(1 - 1)/(-4 - 4) = 0/(-8) = 0

We can reject choice A.

B) (-2, 2)

(2 - 1)/(-2 - 4) = 1/(-6) = -1/6

We can reject choice B.

C) (0, 1)

(1 - 1)/(0 - 4) = 0/(-4) = 0

We can reject choice C.

D. (0, -1)

(-1 - 1)/(0 - 4) = (-2)/(-4) = 1/2

Choice D could be the answer since the slope is 1/2. We could either check whether the line y = 4 - 2x contains the midpoint of RP or whether choice E also has a slope of 1/2. Since it's easier to find the slope, let's do the latter.

E. (2, 0)

(0 - 1)/(2 - 4) = (-1)/(-2) = ½

We see that choice E also has a slope of 1/2. So we need to go back to D and see if the line y = 4 - 2x contains the midpoint of RP.

If D is the correct choice, then the midpoint of RP is ((0 + 4)/2, (-1 + 1)/2) = (2, 0) and we need to check:

0 = 4 - 2(2) ?

0 = 0 (Yes!)

Therefore, D is the correct answer.

Answer: D