Assume Statement 1 alone. Since all three sides of  are congruent - specifically,  - and , it follows by transitivity that . However, no information is given as to whether  has length greater then, equal to, or less than , so which of  and , if either, is the longer cannot be answered.

Assume Statement 2 alone. Since  is the right angle of ,  is the hypotenuse and this the longest side, so  and . However, no comparisons with the sides of  can be made.

Now assume both statements are true.  as a consequence of Statement 1, and  as a consequence of Statement 2, so .

An equilateral triangle has three sides of equal measure, so if the length of any one of the three sides can be determined, the lengths of all three can be as well.

Assume Statement 1 alone.  is a diagonal of a rectangle of area 30. However, neither the length nor the width can be determined, so the length of this segment cannot be determined with certainty.

Assume Statement 2 alone. A square with area 36 has sidelength the square root of this, or 6; its diagonal, which is , has length  times this, or . This is also the length of .

We demonstrate that either statement alone yields sufficient information by noting that the circle that includes all three vertices of a triangle - described in Statement 1 - is its circumscribed circle, and that the circle that includes all three midpoints of the sides of an equilateral triangle - described in Statement 2 - is its inscribed circle. We examine this figure below, which shows the triangle, both circles, and the three altitudes:

The three altitudes intersect at , which divides each altitude into two segments whose lengths have ratio 2:1.  is the center of both the circumscribed circle, whose radius is , and the inscribed circle, whose radius is .

Therefore, from Statement 1 alone and the area formula for a circle, we can find  from the area  of the circumscribed circle:

From Statement 2 alone and the circumference formula for a cicle, we can find  from the circumference  of the inscribed circle:

By symmetry,  is a 30-60-90 triangle, and either way, , and .

An equilateral triangle has three sides of equal length, so  and .

Assume Statement 1 alone. Since , then, by substitution, .

Assume Statement 2 alone. Since , it follows that , and again by substitution, .

Assume Statement 1 alone, and let  be the common perimeter of the triangles. Since an equilateral triangle has three sides of equal length,  and , so .

Assume Statement 2 alone, and let  be the common area of the triangles. Using the area formula for an equilateral triangle, we can note that:

 and ,  

so

.

Assume Statement 1 alone. Since all three sides of  are congruent - specifically,  - and , it follows by transitivity that . However, no information is given as to whether  has length greater than, equal to, or less than , so it cannot be determined which of  and , if either, is the longer. By a similar argument, Statement 2 yields insufficient information.

Now assume both statements are true.  and  are each congruent to one of the congruent sides of equilateral  and are therefore congruent to each other. However, the hypotenuse of a right triangle must be longer than both legs, so the hypotenuse of   is .  is also longer than any segment congruent to one of the legs, which includes all three sides of  - specificially,  is longer than .

Assume Statement 1 alone.  is equilateral, so . Also, by the Triangle Inequality, the sum of the lengths of two sides of a triangle must exceed the third, so . From Statement 1, , so by substitution, , and .

Statement 2 alone provides insufficient information. For example, assume  is an equilateral triangle with sidelength 9. If  is an equilateral triangle with sidelength 8, the conditions of the statement hold, and . However, if  is a right triangle in which , , and , the conditions of the statement still hold, but .

All sides of an equilateral triangle have the same measure, so we can let  be the common sidelength of , and  be that of .

Statement 1 can be rewritten as ; Statement 2 can be rewritten as . The equivalent question is whether we can determine which, if either, is greater,  or . The two statements together are insufficient to answer the question, however; 5 and 10 have sum 15 and product 50, but we cannot determine without further information whether   and , or vice versa. Therefore, we do not know for sure whether a side of  is longer than a side of  - specifically, which of  or  is longer.

All sides of an equilateral triangle have the same measure, so we can let  be the common sidelength of , and  be that of .

Statement 1 can be rewritten as ; Statement 2 can be rewritten as . The equivalent question is whether we can determine which, if either, is greater,  or .

Statement 1 alone yields insufficient information; for example, the two numbers added together could be 10 and 14, but it is impossible to determine whether  or  is the greater of the two. Statement 2 alone is also insufficient, for a similar reason; for example, the two numbers could be 9 and 16, but again, either  or  could be the greater.

Now assume both statements. The only two numbers that can be added to yield a sum of 24 and multiplied to yield a product of 144 are 12 and 12; therefore, , and  and  have the same sidelengths. Specifically,  and  have the same length.

To find an area of a triangle we need the length of the height and the length of the corresponding basis.

Each statement 1 and 2 alone is not sufficient, since we don't know whether the triangle is equilateral. Indeed, we need to take both statements to be able to calculate the area.

Hence, both statements together are sufficient.