Assume Statement 1 alone. The inscribed circle, or "incircle," of a triangle has as its center the mutual point of intersection of the bisectors of the three angles, which, in the case of an equilateral triangle, coincide with the altitudes. , its three altitudes, and the incircle are shown below:

If the area of the incircle is less than , then the upper bound of the radius, which is , can be found as follows:

and  has length less than 4. Also, the point of intersection of the three altitudes divides each altitude into two segments, the ratio of whose lengths is 2 to 1, so 

and 

Therefore, Statement 1 only tells us that , leaving open the possibility that  may be less than, equal to or greater than 10.

Assume Statement 2 alone. The radius of a circle of area  can be found as follows:

The diameter of the circle is twice this, or . Since the longest chords of a circle are its diameters, then any chord in this circle must have length less than or equal to this. Statement 2 tells us that

Now examine the above diagram. , as half of an equilateral triangle, is a 30-60-90 triangle, so by the 30-60-90 Triangle Theorem, 

and 

 is therefore a true statement.

Assume Statement 1 alone. The circumscribed circle, or "circumcircle," of a triangle has as its center the mutual point of intersection of the perpendicular bisectors of the three sides, which, in the case of an equilateral triangle, coincide with the altitudes. , its three altitudes, and the circumcircle are shown below:

The circle has circumference , so its radius, which is equal to the length of , can be found by dividing this by  to yield

.

Also, the point of intersection of the three altitudes divides each altitude into two segments, the ratio of whose lengths is 2 to 1, so 

.

Assume Statement 2 alone. The radius of the circle can be found using the area formula for the circle, and the diameter can be found by doubling this. This diameter, however, only provides an upper bound for the length of a chord of the circle; if  is a chord of this circle, its length cannot be determined, only a range in which its length must fall. Therefore, Statement 2 is insufficient.

Assume both statements are true. From Statement 1 alone,  , and , so  and . Therefore, between  and , two pairs of corresponding sides are congruent. 

 is an equilateral triangle, so ; from Statement 2,  is a right angle, so . This means that the included angle in  is of greater measure, so by the Side-Angle-Side Inequality Theorem, or Hinge Theorem, it has the longer opposite side, or . Both triangles are isosceles, so both altitudes divide the triangles into congruent right triangles, and by congruence,  and  are the midpoints of their respective sides. This means that 

By the Pythagorean Theorem,

and 

Since  and ,

meaning that  is the longer altitude.

Note that this depended on knowing both statements to be true. Statement 1 alone is insufficient, since, for example, had  measured less than , then by the same reasoning,  would have been the shorter altitude. Statement 2 alone is insufficient because it gives information only about one angle, and nothing about any side lengths.

Statement 1 alone is inconclusive, since chords of the same circle can have different lengths.

Statement 2 alone is conclusive. The common side length  of an equilateral triangle depends solely on the area, so it follows that the sides of two triangles of equal area will have the same common side length. Also, each altitude divides its triangle into two 30-60-90 triangles. Examining  and , we can easily find that these triangles are congruent by way of the Angle-Side-Angle. Postulate, so it follows by triangle congruence that .

Let  and  be the common side lengths of   and . The length of an altitude of a triangle is solely a function of its side length, so it follows that the triangle with the greater side length is the one whose altitude is the longer. Therefore, the question is equivalent to which, if either, of  or  is the greater.

Assume Statement 1 alone. This statement can be rewritten as

It follows that  has the greater side length, and, consequently, that its altitude  is longer than .

Assume Statement 2 alone.  divides the triangle into two congruent triangles, so  is the midpoint of ; therefore, . Statement 2 can be rewritten as

This statement is inconclusive. Suppose —that is, each side of  is of length 1. Then , , and  all make that inequality true; without further information, it is therefore unclear whether , the side length of , is less than, equal to, or greater than , the side length of . Consequently, it is not clear which triangle has the longer altitude.

To find the perimeter we should be able to calculate each sides of the triangle.

Statement 1 tells us the area of the triangle. From this we can't calculate anything else, since we don't know whether the triangle is of a special type.

Statement 2 tells us that the triangle is equilateral. Again This information alone is not sufficient.

Taken together these statements allow us to find the sides of the equilateral triangle ABC. Indeed, the area of an equilateral triangle is given by the following formula: . Where  is the area and  the length of the side.

Therefore both statements are sufficient.

To find perimeter, we need the side lengths.

I) Gives us the measure of two angles. The given measurement is equal to 60 degrees. This means the last angle is also 60 degrees.

II) Gives us one side length, but because we know from I) that this is an equilateral triangle, we know that all the sides have the same length. 

Add up all the sides to get the perimeter.

We need I) and II) to find the perimeter

The area of an equilateral triangle is given by the formula

,

where  is its common sidelength. It follows that the triangle with the greater sidelength has the greater area.

We will let  and  stand for the common sidelength of  and , respectively. The question becomes which, if either, of  and  is the greater.

Statement 1 alone can be rewritten by multiplying:

Therefore, .

Therefore, , the length of one side of  is less than , the length of one side of .

Statement 2 alone can be rewritten as . Again, it follows that .

From either statement alone, it follows that .  has the greater sidelength, and, consequently, the greater area.

Since an equilateral triangle has three sides of equal measure, the perimeter of an equilateral triangle is three times its sidelength, so the triangle with the greater common sidelength has the greater perimeter.

Statement 1 gives precisely this information; since one side of  is longer than one side of , it follows that  has the longer perimeter.

Statement 2 gives that  has the greater area. Since the area of an equilateral triangle depends only on the common length of its sides, the triangle with the greater area, , must also have the greater sidelength and, consequently, the greater perimeter.

Assume Statement 1 alone. To find the radius of a circle with area , use the area formula:

This is also the length of each side of , so its perimeter is three times this, or 24.

Assume Statement 2 alone. If we let  be the length of , then, since this the hypotenuse of a 30-60-90 triangle, by the 30-60-90 Theorem, the legs measure  and . Half the product of their lengths is equal to area , so

.

As before, the sidelength of  is 8 and the perimeter is 24.