Statement 1: We can use the area to find the length of square A's side.

  Keep in mind that the ratio of square A to square B is 3:1.

   Now that we know the length of the side, we can find the perimeter of square B.

Statement 2: We can use the diagonal to find the length of the side.

We can easily see the side measures  so we can now find the perimeter.

Recall the formula for perimeter of a square:

where  represents the length of a side. 

Statement 1: We're given  so we can find the perimeter: 

Statement 2: We're given the area  so we can solve for .

With , we can calculate the perimeter: 

Each statement alone is sufficient to answer the question.

Statement 1: We can find the length of the square's side using the information given. 

so 

We can now find the perimeter of the square:  inches

Statement 2: In order to find the length of the square's side using the information provided in Statement 1, we need to use this equation. 

Statement 1 alone is sufficient, but statement 2 alone is not sufficient to answer the question.

Knowing one pair of sides of a quadrilateral to be congruent is not alone sufficient to prove that figure to be a parallelogram, nor is knowing one pair of sides of a quadrilateral to be parallel. But knowing both about the same pair of sides is sufficient by a theorem of parallelograms.

Consider the area formula for a trapezoid:

 is the midsegment of the trapezoid, which, by the Trapezoid Midsegment Theorem, has length equal to the mean of the bases - in other words, . The area formula can be expressed, after substitution, as 

So, if you know both the area and  - but not just one - you can find the height by dividing.

We can work backwards from the perimeter to find the length of the unknown side.

I) Gives us the perimeter.

II) Gives us two of the sides.

In a parallelogram there are two sets of corresponding sides. We can use I) and II) to write the following equaiton, where l is the length of our wanted side.

Solve for l to find our final side:

Statement 1:  A circle with an area of  is enclosed inside the square and touches all four edges of the square.

Write the formula for the area of the circle.

After substituting the value of the area and solve for radius, the radius is 1.  The side length of the square is double the length of the radius.

Statement 2:  A circle with a circumference of  encloses the square and touches all four corners of the square.

Write the formula for the circumference of the circle.

After substituting the value of the circumference and solve for diameter, the diameter of the circle is 1.  This diameter represents the diagonal of the square.  Because the square length and width are equal dimensions, it is possible to solve for the lengths of the square by Pythagorean Theorem knowing just the length of the square diagonal.

Therefore:

Consider isosceles trapezoid .

I)  has a perimeter of .

II) The larger base of  is 45 times bigger than the smaller base.

Find the length of the two legs of .

To find the length of the legs of a trapezoid, we need the perimeter and the two bases. Then, we can subtract the sum of the two bases from the perimeter and divide by two to find the lengths of the sides we are looking for.

Statement I gives us the perimeter of .

Statement II relates the two bases of .

We are told that  is an isosceles trapezoid, which means its two legs are equal; however, we still have too many unknowns and not enough equations. There is no way to solve this without more information.

We still have three unknowns and two equations, so we cannot solve this system of equations.

Each diagonal divides Rhombus  into two triangles, both isosceles.

Statement 1 alone establishes the perimeter of one such triangle. However, it does not make it clear what equal side lengths  and  and diagonal length  are. For example,  fits the perimeter, but so does .

Statement 2 alone gives no information about the actual lengths of the sides.

Assume both statements are true. Since  is equilateral, . It follows that , and . Also, the diagonals of a rhombus bisect their angles and are each other's perpendicular bisectors, so the rhombus, with their diagonals, is given below.

 has perimeter , which means that 

Since  is known to be a  triangle, the proportions of the side lengths are known; along with the above equation, , and, subsequently, the perimeter, can be determined.

To find the perimeter of the quadrilateral, we need to know whether it is of a special type of quadrilaterals and we need to know the length of the sides.

Statement 1 tells us only that the quadrilateral is a rhombus. Indeed, a quadrilateral with perpendicular diagonals intersecting at their midpoint must be a rhombus. However we don't know any length of the sides. 

Statement 2 says gives us the length us two consecutive sides. It could be tempting to answer that it is sufficient, however, we can't conclude that the quadrilateral has equal lengths. Therefore this statement alone is insufficient.

Both statements together are sufficient since we can conclude that the quadrilateral is a rhombus, and twice   will give us the perimeter.