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Exercise:

Construct a right triangle by the hypotenuse and the leg with a compass and a ruler without graduations.


The solution figures:

enter image description here

Note: The text book that I learn math with is on the Ukrainian language, that is the part of cyrillic. So lets rename the figures identifiers below from "б" to "b", from "в" to "c" and "a" is "a", it doesn't need to be changed.


The whole solution by the textbook:

Analysis:
We have two line segments of the length c and b, and b < c (fig. "a"). . Since any hypotenuse is greater than any leg, then our hypotenuse c is the bigger line segment, when the leg b - smaller. So, we need to construct a right triangle ABC in which ∠С = 90°, AB = c, AC = b.

Solution:
Let's draw two perpendicular lines m and n. Let C be their point of the intersection. On the straight line m, we put the segment CA, which is equal to the given leg b (fig. "b"). Let's make a circle with the center at the point A with a radius that is equal to the given hypothesis c. Let this circle cross the line n at two points B₁ and B₂ (fig. "c").
Each of the triangles ACB₁ and ACB₂ are solutions. And as the triangles ACB₁ and ACB₂ are congruent, this problem has a unique solution.

So, why is the last statement true ?
curioushuman
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  • Note that the two triangles are congruent, not just similar. Generally only uniqueness up to congruence is considered. Otherwise you could do the construction again starting at a different point and claim you've found another solution. – eyeballfrog Jul 03 '22 at 15:10
  • Unique upto translations and rotations in the plane I suppose – tryst with freedom Jul 03 '22 at 16:07

2 Answers2

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The problem has a unique solution in the sense that any two such triangles are congruent, independent of how they might have been constructed. That follows from the fact that they have the same three side lengths.

There are many such triangles, because there are many choices for the lines $m$ and $n$. In that sense there is no "unique solution".

I think the text is confusing when it calls attention to the fact that the proposed construction happens to find two instances of the required triangle.

Ethan Bolker
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  • Thank you for your answer, but I'm still confused, what that means. And also, why couldn't the solution have been found by a single triangle, because, for example *ACB₁*, is the triangle we needed to find, so why do I even need the second triangle ? – curioushuman Jul 03 '22 at 16:09
  • You don't "need" the second triangle, but the construction produces it since the circle intersects $n$ at two points. The text then tries to explain why this does not matter. I think the text is badly written (or badly translated here), which is why you are confused. – Ethan Bolker Jul 03 '22 at 16:26
  • I've translated the whole text by myself, it wasn't performed by a translator. But even though, I've read this solution for 15 times, probably, I still do not understand why the book added the second triangle if the answer could be just some of them. And what means: "And as the triangles ACB₁ and ACB₂ are congruent, this problem has a unique solution." I still don't get. Either I'm dumb either something else. – curioushuman Jul 03 '22 at 18:47
  • Your understanding is fine. The book's author should have drawn just one of the triangles and omitted that last sentence, or replaced it with a complete discussion of what "unique solution" means. As it stands it's just a stumbling block for a careful reader like you. Move on and don't worry. – Ethan Bolker Jul 03 '22 at 18:55
  • Thank you, Ethan ! – curioushuman Jul 04 '22 at 18:46
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The same triangle sitting on another place on your paper is as a triangle the same. you could have many different triangles , if you give only two sides and no angle. You have two realy different triangles if you are given two adjacent sides and the angle oposit

trula
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