If, then

.

This makesan equiangular triangle.

If, andis equiangular, then, since corresponding angles of similar triangles are congruent,has the same angle measures, and is itself equiangular.

From either statement, since all equiangular triangles are equilateral, we can draw this conclusion about.

The two statements together provide insufficient information. A triangle with sides,, andis equilateral and has perimeter;一个三角形with sides,, andis not equilateral and has perimeter.

To find the answer we should know more about the characteristics of the triangle, i.e. its angles, sides...

Statement 1 alone is obviously insufficient, since we don't know whether the triangle is equilateral, nothing can be said about AB.

Statement 2 is equally as unhelpful as statement 1, since we don't know whether ABC is of a specific type of triangle.

Taken together, these statements allow us to calculate the length of CB, but we can't go further, because we don't know what is AD.

Therefore statements 1 and 2 are not sufficient even taken together.

To find the length of the side, we would need to know anything about the lengths in the circle or in the triangle.

From statement 1, we can find the radius of the circle, which allows us to calculate the height of the triangle, since the radius is的高度。最后由于三角形是equilateral, we can also calculate the length of the sides from the height.

Therefore statement 1 is sufficient.

Statement 2 also gives us useful information, indeed the perimeter is simply three times the length of the sides.

Therefore the final answer is each statement alone is sufficient.

I) Tells us the perimeter of the triangle.

II) Tells us that FHT is an equilateral triangle.

Taking these statements together we are able to find the side length by dividing the perimeter from statement I, by 3 since all side lengths of an equilateral are the same by statement II.

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

Assume Statement 2 alone. Sinceis the right angle of,is the hypotenuse and this the longest side, soand. However, no comparisons with the sides ofcan be made.

Now assume both statements are true.as a consequence of Statement 1, andas 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 lengthtimes 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 findfrom the areaof the circumscribed circle:

From Statement 2 alone and the circumference formula for a cicle, we can findfrom the circumferenceof the inscribed circle:

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

An equilateral triangle has three sides of equal length, soand.

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 letbe the common perimeter of the triangles. Since an equilateral triangle has three sides of equal length,and, so.

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

and,

so

.