What are the Coordinates of the Foot of the Perpendicular from the Point GMAT Problem Solving

Question: What are the coordinates of the foot of the perpendicular from the point (2,2) to the line y - 2x - 8 =0?

1. ( 4 ,-2)
2. (-4,2)
3. (2,4)
4. (2,-4)
5. (-2,4)

Correct Answer: E
Solution and Explanation:
Approach Solution 1:

The problem statement asks to find out the coordinates of the foot of the perpendicular from the point (2,2) to the line y-2x-8=0.

y-2x-8=0
y=2x+8
Therefore, the slope of this line= 2

The line including points (2,2) is perpendicular to this line. Therefore, the product of the slopes of both these lines will be -1.
The slope of the line perpendicular to y-2x-8=0 is -1/2

Point (2,2) lies on this perpendicular line.
Therefore, the equation of the new line,
y-2 = (-1/2)(x-2)
2y-4 = -x+2
2y = -x+6
Therefore only the point (-2,4) fulfills this equation.

Hence, the coordinates of the foot of the perpendicular from the point (2,2) to the line y-2x-8=0 = (-2,4)

Approach Solution 2:

The problem statement asks to find out the coordinates of the foot of the perpendicular from the point (2,2) to the line y-2x-8=0.

y-2x-8=0
The Slope intercept form is therefore: y =2x+8
The slope of this line= 2

The foot of the perpendicular and the point (2,2) will make a line perpendicular to the given line.
Therefore, the slope will be -1/2 ( since the product of slope of perpendicular lines = -1)

Let’s solve the problem by analysing the given options:

1. (4,-2) & (2,2): Slope = (2+2)/(2-4)= -2
2. (-4,2) & (2,2): Slope = (2-2)/(2+4) =0 (since y coordinate is the same for two points, the line is horizontal)
3. (2,4) & (2,2): Slope = (2-4)/(2-2) = Not defined (since x coordinate is the same for two points and the line is Vertical)
4. (2,-4) & (2,2): Slope = (2+4)/(2-2) = Not defined (since x coordinate is the same for two points and the line is Vertical)
5. (-2,4) & (2,2): Slope = (2-4)/(2+2)= -1/2

Therefore only option E (-2,4) satisfies the question.
Hence, the coordinates of the foot of the perpendicular from the point (2,2) to the line y-2x-8=0 = (-2,4)

Approach Solution 3:

The problem statement asks to find out the coordinates of the foot of the perpendicular from the point (2,2) to the line y-2x-8=0.

The given line is y=2x+8
Hence, the slope of the perpendicular is -1/2

Let assume the equation is y = -(1/2)x + c
Since this line passes through (2,2), let’s put the values to get c
2 = -(1/2)*2 + c
Therefore, c = 3

Hence, the equation of the perpendicular is y= -(1/2)x + 3

The question asks for the foot of the perpendicular on the given line. This implies we need to find the point that lies on both the line and the perpendicular. At that point, the y coordinate has to be similar for both lines.
Therefore, we get
2x+8= -(1/2)x + 3
By solving, we get, (5/2)x= -x = -2

Hence we are left with the choice E which has -2 as the x coordinate.
Therefore, the coordinates of the foot of the perpendicular from the point (2,2) to the line y-2x-8=0 = (-2,4)

“What are the coordinates of the foot of the perpendicular from the point”- is a topic of the GMAT Quantitative reasoning section of the GMAT exam. This topic has been taken from the book “Kaplan GMAT Math Workbook”. The candidates can improve their knowledge of mathematics by solving more GMAT Problem Solving questions. The candidates can practice from the GMAT Quant practice papers to be familiar with varieties of questions that will help them to score better in the exam.

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