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# Triangle with a given perimeter

## Question

You are given a point M and an angle C in the plane. Using a ruler and a compass, draw a line through the point M that cuts from the angle C a triangle with a given perimeter p.

In the figure below, a green line has been drawn through the point M that creates a green triangle with the angle C. The task is to create such a triangle that will have a given perimeter p. The ruler you are given functions as a straight edge and also allows you to measure distances.

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Edited by bonanova
Gold Star
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measure MC length

measure the 2 angles formed with vertex C

apply sign law for triangle, calculator is needed.

4 unknown lengths, compute angle form with vertex M

Use compass and straight edge to make that angle on the ground

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For field work using Meter tape
as compass and ruler.

given length p..
make the cutting lines from M
let intersection points be c1 & c2 )
measure C-c1 , C-c2 & c1-c2
add and compare to p..

adjust c2 along its line
measure C-c1 , C-c2 & c1-c2
add and compare to p..

repeat until distance sum=p is ok

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no clue on this one, you can divide a single line segment to any measurement, but showing any point m to construct a triangle of any parameter? much much harder i think.

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no clue on this one, you can divide a single line segment to any measurement, but showing any point m to construct a triangle of any parameter? much much harder i think.

What are you able to do with a compass? How might that be of use here?

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For field work using Meter tape
as compass and ruler.

given length p..
make the cutting lines from M
let intersection points be c1 & c2 )
measure C-c1 , C-c2 & c1-c2
add and compare to p..

adjust c2 along its line
measure C-c1 , C-c2 & c1-c2
add and compare to p..

repeat until distance sum=p is ok

A great engineering solution to the puzzle. Thanks.

Suppose the ruler and compass were exact and precise instruments.

Could you then construct the triangle exactly, on the first try?

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measure MC length

measure the 2 angles formed with vertex C

apply sign law for triangle, calculator is needed.

4 unknown lengths, compute angle form with vertex M

Use compass and straight edge to make that angle on the ground

can you do it using only geometry?

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The solution requires only two steps. I've alluded to the probability that the first step involves using the compass.

The puzzle has an interesting background: It is one of a group of problems that, for discriminatory reasons, was crafted to be very difficult to solve, but that nevertheless had a simple solution. In that sense it is an Aha! type of a problem. It is not at all clear how to proceed; but the solution, once seen, is, like, oh yeah ... but I never would have thought of it. Since that's probably the case now, I will not give the answer but I will allude to the first step, leaving out some needed information. If you read the clue, the problem will be simply to figure out the needed information and use it. The second step will then be easily found.

OK, actually, the first statement in this post might now be enough of a first clue. The clue described in the paragraph will follow, if needed.

I will of course award the coveted bonanova Gold Star for a solution.

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beats me..

Looks like a very fine puzzle!

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Here's the clue, then.

The first step is to draw a circle inside the angle and tangent to both edges of the angle. That reduces the puzzle to determining where the points of tangency should be, and what to do next.

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As promised ...

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