Question

Find the coordinates of the foot of the perpendicular from point G(-2,-4) to the line joining points M(0,2) and P(-1,4).

A) (1,3)
B) (-1,1)
C) (-2,0)
D) (0,2)

Answer 1

The foot of the perpendicular from G(-2,-4) to the line MP is at the point (2, -2). This was found by first determining the equation of line MP, then finding a perpendicular line through G, and finally solving for their intersection, which does not match any of the provided choices.

Step-by-step explanation:

To find the coordinates of the foot of the perpendicular from point G(-2,-4) to the line joining points M(0,2) and P(-1,4), we must first determine the equation of the line MP and then find the point at which a perpendicular line through G will intersect MP.

The slope of line MP is given by: (4 - 2) / (-1 - 0) = 2 / -1 = -2. The equation of line MP can be written in point-slope form as y - 2 = -2(x - 0), which simplifies to y = -2x + 2. To find a line perpendicular to this one, we need a slope that is the negative reciprocal, which is 1/2. The equation of the line perpendicular to MP that passes through G is (y - (-4)) = 1/2(x - (-2)), or y = 1/2x - 3.

Now, we have to find the intersection of the two lines by solving for x and y:

• y = -2x + 2
• y = 1/2x - 3

Setting the two equations equal to each other:

-2x + 2 = 1/2x - 3

Multiplying both sides by 2 to clear the fraction gives:

-4x + 4 = x - 6

Adding 4x to both sides:

4 = 5x - 6

Adding 6 to both sides:

10 = 5x

Dividing both sides by 5:

x = 2

Plugging x = 2 into the first equation:

y = -2(2) + 2 = -4 + 2 = -2

So, the coordinates of the foot of the perpendicular from G to line MP are (2, -2), which is not listed in the options provided, indicating a possible typo or error in the question or the answer choices.