For a given right triangle with a side length and a height , the area is

. Plugging in the values provided:

We can view the horizontal segment connecting , and as the base; its length wiill be . The height will be the perpendicular (vertical) distance to this segment from the opposite point , which is , the -coordinate; therefore, the area of the triangle will be half the product of these two numbers, or

.

The vertical segment connecting and can be seen as the base of this triangle; this base has length . The height is the perpendicular (horizontal) distance from to this segment, which is 6, the same as the -coordinate of this point. The area of the triangle is therefore

.

Since are right triangles, we know that is an isosceles right triangle. So we know that the lengths of and are 2 cm, so we can get the length of by using the Pythagorean Theorem:

is the midpoint of , so the length of is .

Now we can use the Pythagorean Theorem again to solve for : .

Finally, we have all the elements needed to solve for the area of :

We are looking for ways to add three primes to yield a sum of 43. Two or all three (since an equilateral triangle is considered isosceles) must be equal (although, since 43 is not a multiple of three, only two can be equal).

We will set the shared sidelength of the congruent sides to each prime number in turn up to 19:

By the Triangle Inequality, the sum of the lengths of the shortest two sides must exceed that of the greatest. We can therefore eliminate the first three. , , and include numbers that are not prime (21, 15, 9). This leaves us with only one possibility:

- greatest length 19

19 is the correct choice.

A scalene triangle has three sides of different lengths, so we are looking for three distinct prime integers whose sum is a prime integer.

One of the sides cannot be 2, since the sum of 2 and two odd primes would be an even number greater than 2, a composite number. Therefore, beginning with the least three odd primes, add increasing triples of distinct prime numbers, as follows, until a solution presents itself:

- incorrect

- correct

The correct answer, 19, presents itself quickly.

A scalene triangle has three sides of different lengths, so we are looking for three distinct prime integers whose sum is 47.

There are ten ways to add three distinct primes to yield sum 47:

By the Triangle Inequality, the sum of the lengths of the shortest two sides must exceed that of the greatest. We can therefore eliminate all but four:

The greatest possible length of the longest side is 23.

The three sides of a scalene triangle have different measures. One measure cannot have is 12, but this is not a choice.

It cannot be true that . Since the perimeter is

, we can find out what other value can be eliminated as follows:

Therefore, if , then , and the triangle is not scalene. 9 is the correct choice.

In an isosceles triangle, at least two angles measure the same. Therefore, one of three things happens:

Case 1: The two given angles have the same measure.

The angle measures are , making the triangle equianglular and, subsequently, equilateral. An equilateral triangle is considered isosceles, so this is a possible scenario.

Case 2: The third angle has measure .

Then, since the sum of the angle measures is 180,

as before

Case 3: The third angle has measure

as before.

Thus, the only possible value of is 40.

For a triangle, the sum of the two shortest sides must be greater than that of the longest. We are given two sides as 5 and 12. Our third side must be greater than 7, since if it were smaller than that we would have where is the unknown side. It must also be smaller than 17 since were it larger, we would have .

Thus our perimeter will be between and . Only II and III are in this range.