A Certain Computer Program Randomly Generates Equation Of Line Is Form GMAT Problem Solving

Question:

A certain computer program randomly generates equation of line is form of y=mx+b.If point p is a point on a line generated by this prog, what is probability that line does not pass through ABCD.

1. 3/4
2. 3/5
3. 1/2
4. 2/5
5. ¼

“A certain computer program randomly generates equation of line is form” - is a topic of the GMAT Quantitative reasoning section of GMAT. To solve GMAT Problem Solving questions a student must have knowledge about a good number of qualitative skills. GMAT Quant section consists of 31 questions in total. The GMAT quant topics in the problem-solving part require calculative mathematical problems that should be solved with proper mathematical knowledge.

Approach Solution : 1

 

Any line passing through the x-axis point P at 6 will go through one of the four equally sized dotted regions (except x axis).

Lines passing through two of these regions are unacceptable (represented by red lines), while lines passing through the other two are acceptable (green lines).

Therefore, 1/2 of the area is suitable.

There is a 1/2 chance that the line will not pass through ABCD.

Correct Answer: (C)

Approach Solution : 2

If the original image is anything to go by, the coordinates of the points C, D, and P are such that the angle DPC is 90. (This can be confirmed using slopes or coordinates.)

Therefore, for any line passing through point P and having the formula y=mx+b, it cannot pass through a 90' region. This is because doing so would cause it to enter the ABCD square. It is free to pass wherever it wants outside of this 90. It'll be alright,

This line can be drawn in a 180' region that extends from the x-axis in an anticlockwise direction. (Only taking into account above x-axis; once you rotate the line sufficiently to go below x-axis, the opposite end would be above x-axis)

Probability is therefore equal to the product of the total favorable area and the total area which gets the value 1/2 .

Now let's talk about how 90' and 180' work. Only the first quadrant or the first and fourth quadrants together are viable options for this solution. However, when we take into account both quadrants, it will offer a mirror image along the x-axis.

Total area formed by line = 90*2, which is not desired, and total area that can be formed by line = 180*2, which is 360

The answer is still the same, which is 1/2.

Correct Answer: (C)

Approach Solution : 3

Make O the point on CD that forms a right angle with P

Using the Pythagorean theorem on the POC and POD triangles. We now can see that PC = PD

Therefore, since PCD is an isosceles triangle, Angle CPD = 90

Therefore, the line will cross the rectangle if 0< = Angle CPD <= 90

Favorable outcomes therefore equal 90, while potential outcomes equal 180

Therefore, the likelihood will be 90/180 = ½

Line not crossing through the rectangle has a probability of 1 - 1/2 = 1/2

Correct Answer: (C)

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