Answer:
Triangles I and III are right triangles, because they are similar to a 3-4-5 right triangle.
Explanation:
The Pythagorean theorem tells you of the relationship between the sides of a right triangle. A set of integer side lengths that satisfies the Pythagorean theorem is called a Pythagorean triple. There are an infinite number of these, and there are formulas for finding them.
The smallest Pythagorean triple is {3, 4, 5}, and it is generally the first one you learn about. It is special for a variety of reasons. One of them is that it is the only Pythagorean triple that is an arithmetic sequence. Any multiple of a Pythagorean triple is also a set of side lengths that will form a right triangle.
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If we multiply the {3, 4, 5} triple by 2/10, we get {6/10, 8/10, 1}, which matches the side lengths of Triangle I in your list. Multiplying by 1/13 gives {3/13, 4/13, 5/13}, which matches the side lengths of Triangle III in your list. That means both Triangles I and III are right triangles.
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The Pythagorean theorem tells you that a right triangle with sides a, b, and hypotenuse c will satisfy the relation ...
c² = a² +b²
If the triangle is isosceles, such that a=b, then the hypotenuse will be ...
c² = a² +a² = 2a²
c = a√2
In reduced terms, the sides of an isosceles right triangle have the ratios ...
{1, 1, √2}
That is, there is no rational Pythagorean triple that describes an isosceles triangle. Triangle II cannot be a right triangle.
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Additional comment
Just so that you can see {3, 4, 5} is a Pythagorean triple, we show you ...
5² = 3² +4² ⇔ 25 = 9 + 16
Other small integer triples that may be seen in math problems are ...
{5, 12, 13}, {7, 24, 25}, {8, 15, 17}, {9, 40, 41}, {11, 60, 61}, {12, 35, 37}