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.

Knowing that the triangle has a right angle indicates the question can be solved using the Pythagorean Theorem, however it is unclear which side is the hypotenuse. For example, if 8 is not the hypotenuse, the length of the third side is 10. If 8 is the hypotenuse, the length of the third side is 5.3.

Additionally, if you only know that 8 is the longest side, the length of the third side could be anything greater than 2 and less than 8. Therefore, having both pieces of data will allow you to solve the problem.

The side opposite the angle of greatest measure is the longest of the three, so if we can determine which angle is the longest, we can answer this question.

It follows from Statement 1 by definition that , so, since the measures of the three angles total , , making  right and the other two acute. This proves  has the greatest measure of the three.

It follows from Statement 2 that  is either right or obtuse; therefore, . Subsequently, the other two angles are acute, so again,  has the greatest measure of the three.

From either statement alone, we can therefore identify  as the side of greatest measure.