Assume Statement 1 only. By the Pythagorean Theorem, , so ; subsequently, , and one side congruence is proved. However, this, along with one angle congruence - the congruence of right angles  and  - is not enough to prove or disprove triangle congruence.

Assume Statement 2 only. By the Pythagorean Theorem, . Since , it follows that , and, subsequently, . Since, if , , it follows by contradiction that  is a false statement.

Assume Statement 1 alone is true. Since each right triangle is inscribed inside a circle, each of the right angles is inscribed in that circle, and each intercepts a semicircle. That makes the two hypotenuses of the triangles,  and , diameters; since they are on the same circle, their lengths are the same. However, no information is given about any of the other sides or angles, so no congruence can be proved or disproved.

Statement 2 alone gives us only congruence between one set of corresponding legs, which, along with one angle congruence, is insufficient to prove or to disprove triangler congruence.

Now assume both statements are true. From Statement 1, it follows that congruence of hypotenuses, and Statement 2 gives us congruence of corresponding legs; this sets up the conditions of the Hypotenuse-Leg Theorem, so it follows that .

Assume both statements are true. We show that both statements are insufficient to answer the question.

Suppose  and . 

 and , satisfying the conditions of Statement 2.

The area of a right triangle is half the product of the lengths of its legs. Each triangle therefore has as its area , making the areas the same; Statement 1 is satisfied.

Since corresponding legs of the triangles are congruent,  by the Side-Angle-Side Postulate.

Now, suppose  and . 

 and , satisfying the conditions of Statement 2.

The area of a right triangle is half the product of the lengths of its legs. Each triangle therefore has as its area , making the areas the same; Statement 1 is satisfied.

However,  and , so it is not true that .

Assume Statement 1 alone. The statement gives that all four legs of both triangles are congruent - specifically,  and . Since the right angles are also congruent, then, by the Side-Angle-Side Postulate, .

Assume Statement 2 alone. The statement gives that all four acute angles are congruent - specifically, that  and . However, since we do not have any congruence or noncongruence between corresponding sides, congruence of the triangles cannot be proved or disproved.

In order to determine how far apart the ships are now, it is necessary to know how far each ship is from New York now. Even knowing both statements, however, you only know that the paths are at right angles, and that the ratio of the two distances is 6 to 5. Without knowing the time elapsed, which is not given, you cannot tell how far apart the ships are.

The answer is that both statements together are insufficent to answer the question.

We need both statements here because we need to work with ratios.

If you notice that a 12/16/20 triangle follows the 3/4/5 triangle ratio, then it is even easier. Otherwise, you can set up the following proportion and solve to find the answer.

Where x is the hypotenuse of triangle HJK.

To find the hypotenuse of a right triangle we can use the classic Pythagorean Theorem. To do so, however, we need to know the other two sides.

I) Tells us how to relate the other two sides to eachother.

II) Gives us the length of one side.

Use II) and I) to find the two short sides. Then use Pythagorean Theorem to find the perimeter.

, where SS represents the shortest side, MS represents the middle side, and H represents the hypotenuse.

Statement 1:In order to find the hypotenuse, we need to find the values for the base and length. Although we're given the area, we need additional information. 

but the base and height can be 2 and 18, 3 and 12, 4 and 9, or 6 and 6.

Statement 2: We don't need this information to find the length of the hypotenuse.

Statements 1 and 2 are not sufficient, and additional data is needed to answer the question.

Statement 1: We're given the area but require additional information.

The base and height, however, can be 2 and 15, 3 and 10, or 5 and 6.

Statement 2: We're provided the information that narrows our possible values. The difference between the base and height is  so our values must be 5 and 6. 

Using BOTH statements, we can find the hypotenuse of the right triangle via the Pythagorean Theorem:

Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient.

Statement 1: We'll need more information to answer the question.

Which means the base and height can either be 4 and 3, or 6 and 2. We cannot find the length of the hypotenuse unless we know which one.

Statement 2: We're given the lengths of the base and height so we can just plug them into the Pythagorean Theorem.