Sloppy Products

Geometry Level 2

The circle below is centered at $O$ and has a radius of 2.

The 144 lines passing through the origin divide the quarter-circle in the first quadrant in such a way that each of the resulting 145 sectors has the same area.

Determine the product of the slopes of all these 144 lines.

The answer is 1.

2 solutions

Maninder Dhanauta
Aug 16, 2016

It is observed that the angle between the lines would be $\frac{90}{145}$

Let $\frac{90}{145}$ = $\theta$

Since the slope is basically the $tan \theta$ , the product can be written as :

$tan(\theta) tan(2\theta) tan(3\theta) \ldots tan(90-2\theta) tan(90-\theta)$

And since the tan of complement angles are reciprocals, meaning:

$tan(\alpha)* tan(90-\alpha) = 1$ ,

everything boils down to 1

Note: If it were 145 lines(an odd number) instead of 144 lines(an even number)

the answer would have still been the same ( tan(45)*tan(90-45)=1 )

Moderator note:

Yes. Essentially, this boils down to showing that the product of pairs of slopes are all 1. And so the product in question is equal to $1\times 1\times \cdots 1 = 1$ as well.

There's another way to phrase your answer.

Hint :

Let $m_1 < m_2 < m_3 < \cdots < m_{144}$ denote the slopes of these straight lines, it remains to show that $m_n \times m_{145 - n} = 1$ , where $n = 1,2,3,\ldots , 144$ .

By constructing a right triangle with an interior non-right angle $\theta$ , then we have $\tan \theta \tan(90^\circ - \theta) = 1$ . What is the relationship between this equation and $m_n \times m_{145 - n} = 1$ ?

Now we want to evaluate $\displaystyle\prod_{n=1}^{144} m_n = m_1 \times m_2 \times \cdots \times m_{144} = (m_1 \times m_{144}) \times (m_2 \times m_{143}) \times \cdots \times (m_{72} \times m_{73})$ .

In response to Challenge Master's Note:

First, a line with slope 1 (or angle of 45 degree) will divide the 1st quadrant such that both halves are symmetrical.

Second, what is slope? It is (y1-y2)/(x1-x2), the rate of change per 1 unit along the x-axis.

Because of the symmetry, there should also be a line with the same rate of change but, this time, along the y-axis.

(Because we already know that the angles between the corresponding lines and axes is the same, therefore the tans are "similar" in a sense) Meaning, slope =(x1-x2)/(y1-y2)

As you can see, these are reciprocals and will multiply to equal 1.

* Interesting fact: These 2 lines are inverse to each other (reflect over the line y=x). The slopes of 2 such lines are reciprocals. *

Maninder Dhanauta - 4 years, 9 months ago

The way the problem is worded is a bit ambiguous. By your solution, you mean that the angle between successive lines is $\frac{90}{144}$ . But even if we just look at the first quadrant (it's not even clear that all of the 144 lines are confined to the 1st & 3rd quadrants), $143$ lines divide it into $144$ parts, and then what about lines coinciding with either the $x$ or $y$ axes? Any of $143, 144, 145$ lines can divide the 1st quadrant into $144$ equal parts.

I think you should make it clear that it is the first quadrant that is divided into $144$ equal parts.

Michael Mendrin - 4 years, 10 months ago

Thanks for bringing that to my attention. I meant to make it 144 distinct lines that divide the quarter circle into 145 sectors of equal area. I have fixed the solution and have written 90/145. I meant to make the solver think a little that's why i didn't state the number of sectors clearly.

Maninder Dhanauta - 4 years, 10 months ago

Tina Sobo
Sep 27, 2016

Number the lines from 1 to 144, starting with the closest line to the x-axis. Then consider the lines in pairs, letting a=the angle formed by the first line to the x-axis, which equals the angle formed between any two consecutive lines, since the quadrant is divided equally.

Slope of line 1 = sin(a)/cos(a) Slope of line 144 =cos(a)/sin(a) So the product of slope(line 1) * slope(line 2) = 1

Slope of line 2 = sin(2a)/cos(2a) Slope of line 143 = cos(2a)/sin(2a) Product of pair = 1

Likewise, for each pair of lines moving inward, the product of their slopes will be 1. Since there are an even number of lines, the product is 1. Incidentally there were an odd number of lines, the product would still be one, since the middle line would bisect the original quadrant, making a 45-degree angle where sin(45)=cos(45), so sin(45)/cos(45) = 1.

Sloppy Products

The answer is 1.

2 solutions

Moderator note:

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