This is my first time to encounter trapezints, so I only depend on your posted desciptions re trapezint:

"Trapezints are those special trapezia whose sides have integer lengths. The parallel sides are of unequal length."

The reading is from the top parallel side, then clockwise.

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a. What is the shorter perimeter that a trapezint can have? Explain why there is not a smaller one.

You mean "shortest", instead of the posted "shorter"?

I assume "shortest, so,

Let us start with the smallest positive integer, 1. So the shorter of the parallel sides is 1.

The next smallest positive integer is 2. Then, the other parallel side is 2.

For the other two sides, the two legs, we use 1 for each.

Hence this trapezint is [1,1,2,1].

It is an isosceles trapezint.

Its perimeter is 5. It is the shortest perimeter that a trapezint can have.

There is none smaller than 5 perimeter because there is no positive integer that is lower than 2. This side 2 cannot be lower than 2. Otherwise, if it could be lower, then it could be 1 only, and the perimeter is 4 only. But then the two parallel sides are equal. That cannot be, because it is said that the two parallel sides must be of unequal lengths.

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b. What is the smallest perimeter of a trapezint with a non-parallel sides having different lengths? Explain why there is not a smaller one.

That means all 4 sides are unequal in lengths.

The shortest distance between two parallel lines is the perpendicular distance between them.

The right triangle with all sides as integers that has the shortest sum of the 3 sides is the 3-4-5 triangle.

Using those, here is the trapezint for this part (b).

The top parallel side is 1.

The right leg is 3. It is perpendicular to the top and bottom sides. It is the shorter leg of the 3-4-5 right triangle.

The bottom parallel side is 1+4 = 5.

The left leg is 5. It is the hypotenuse of the 3-4-5 triangle.

In short, the trapezint is [1,3,5,5].

Its perimeter is 14.

Another trapezint for this part (b) is [1,4,4,5], whose perimeter is also 14.

This is the same as the [1,3,5,5], except that here, the leg 4 of the 3-4-5 right triangle is the perpendicular distance between the top and bottom parallel sides.

Why there is no trapezint for this part (b) that has a perimeter lower than 14? Because there is no right triangle of integer sides that is smaller than the 3-4-5 triangle.

These special right triangles are Pythagorean Triples: 3-4-5, 5-12-13, etc.

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c. What is the smallest perimeter of a trapezint with at least one angle a right angle? Explain why there is not a smaller one.

My aswer here is exactly the same as those in part (b) above: [1,3,5,5] or [1,4,4,5].

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d. Find all trapezints with perimeter 9.

Note: here we regard two trapezints as different only if they are not congrunt. In particular [a,b,c,d] and [a,d,c,b] are the same.

The trapezints here are all isosceles trapezints, meaning, the non-parallel legs are equal in lengths.

They are:

>>>If the top parallel side is 1:

[1,1,6,1], [1,2,4,2], [1,3,2,3]

>>>and if the top parallel side is 2:

[2,1,5,1], [2,2,3,2]

>>>and if the top parallel side is 3:

[3,1,4,1]

Meaning, I found 6 trapezints whose perimeters are 9 each.