Quadrilaterals - GMAT Quantitative

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.

Because all 4 sides of a square are equal, knowing the length of one side is sufficient to answer the question. Using the Pythagorean Theorem, you can calculate the length of 1 side of a square by knowing the length of a diagonal and then calculate the area.

The area of the black region is one-fourth the difference of the areas of the circle and the square.

If is the sidelength of the square, then the length of its diagonal - which is also the diameter of the circle - is, by the Pythagorean Theorem, , and the radius . Therefore, if you calculate either the radius or the sidelength, you can calculate the other, allowing you to find the areas of the circle and the square.

Statement 1 allows you to find the sidelength; just divide 40 by 4.

Statement 2 allows you to find the radius; just solve for in the equation .

Therefore, either one gives you enough information to solve the problem.

The diameter of the inscribed circle is equal to the the sidelength of the square. From Statement 1, the circumference can be divided by to obtain this measure, and this can be squared to obtain the area of the square.

Statement 2 gives extraneous information, as, by regularity of the figure, it is already known that is one fourth of the circle and, subsequently, a arc.

From Statement 1, the radius of the circle can be calculated by working backwards from the area formula, and the diameter can be calculated by doubling this.

From Statement 2, since, by the regularity of the square, is one fourth of the circle, the length of can be multiplied by four to get the circumference of the circle. This can be divided by to obtain this diameter.

This diameter is equal to the the sidelength of the square in which it is inscribed, so it can be squared to obtain the area of the square. This makes each statement alone sufficient to answer the question.

You can use the diagonal of a square to find its side length via the ratio of a 45/45/90 triangle.

In this case our triangle will have side lengths of 15.

We can also divide our perimeter by 4 to get our side length, which is again 15.

Therefore, each statement alone is sufficient to solve the question.

Assume Statement 1 alone. The perimeter of a square is four times the length of a side, which here is ; the perimeter is greater than 100, so

Therefore, the question can be answered in the affirmative from Statement 1 alone.

Statement 2 alone gives insufficient information, however. The area of a square is the square of the length of a side. If the sidelength is , for example, the area is the square of this, which is 81; if the sidelength is , the area is 121. Both areas exceed 40, but only in the second case is .

We can use the side length and the Pythagorean Theorem to find the diagonal of a square.

We can find side length from area, so we could solve this with either I or II.

The diagonal of the square can be calculated as long as we have any information about the lengths or area of the circle or of the square.

Statement 1, by giving us the area of the circle, allows us to find the radius of the circle, which is half the length of the side. Therefore statement 1 alone is sufficient.

Statement 2, by telling us the length of a side of the square is also sufficient, and would allow us to calculate the length of the diagonal.

Therefore, each statement alone is sufficient.

We are asked to find the length of a diagonal of a square.

We can do this if we have the side length. We can find side length from either perimeter or area.

From Statement I)

In this case, our side length is 15 meters.

We can use this and Pythagorean Theorem or 45/45/90 triangles to find our diagonal.

From Statement II)

From here, we can plug the side length into the Pythagorean Theorem like before and solve for the diagonal.

Therefore, either statement alone is sufficient to answer the question.

The length of the diagonal of a square is given by , where represents the square's side. As such, we need the length of the square's side.

Statement 1:

Statement 2:

Both statements provide us with the length of the square's side.

Statement 1: The information provided would only be useful if the ratio of square A to square B was known.

Statement 2: We need the length of the square's side to find the length of the diagonal and we can use the area to solve for the length of the side.

Now we can find the diagonal:

To solve this problem, we must recognize that the diagonal bisector creates identical 45˚ - 45˚ - 90˚ right triangles. This means that, if the sides of the square are then the diagonal must be . We can then set up the following equation:

If then the area must be:

A rectangle, by definition, is a parallelogram. Statement 1 asserts that the diagonals of this parallelogram are perpendicular. Statement 2 asserts that adjacent sides of the parallelogram are congruent, so, since opposite sides are also congruent, this makes all four sides congruent. From either statement alone, it can be deduced that Rectangle is a rhombus. A figure that is a rectangle and a rhombus is by definition a square.

Consider the following equations:

Where a is area, p is perimeter, and s is side length

We can find the side length with either our area or our perimeter.

Thus, we only need one statment or the other.

Since ABCD is a square, we just need to know the length of the diagonale to find the length of the side. BE is half the diagonal, therefore knowing its length would help us find the length of the sides.

Statement 1 tells us the length of BE, therefore, with the formula where is the diagonal and the length of side, we can find the length of the side.

Statement 2 tells us that triangle AEB is isoceles, but it is something we could already have known from the beginning since we are told that E is the midpoint of the diagonal.

Therefore, statement 1 alone is sufficient.

To find the area of a square we need to find its side length.

In a square, the diagonal allows us to find the other two sides. The diagonal of a square creates two 45/45/90 triangles with special side length ratios.

I) Gives us the diagonal, which we can use to find the side length, which will then help us find the area.

II) Perimeter of a square allows us to find side length, which in turn lets us find area.

So, either statement is sufficient.

Statement 1) mentions that the area of a quadrilateral is 4. This statement is insufficient to solve for the length of the square because the family of quadrilaterals include any 4-sided shape with 4 interior angles. Examples of quadrilaterals are squares, rectangles, rhombus, and trapezoids, but the quadrilateral is not necessarily a square.

Statement 2) mentions that all four interior angles of a quadrilateral are right angles. This narrows down the shape to either a square or a rectangle. Both shapes have 4 right angles, but there is not enough information to determine if the shape is a square or a rectangle.

Therefore, neither statement is sufficient to solve for the length of a quadrilateral.

Statement 1) gives the area of the square. For all positive real numbers, the formula, , or , can be used to find either area or side length interchangeably.

Statement 2) mentions that the diagonal is 1, which is a positive real number. The formula , can be used to also find the side length.

Either statement alone is sufficient to solve for the length of the square.

We are asked to find the perimeter of a square. To do that we need a side length.

I) Is irrelevant and is trying to distract you with other aspects of the question.

II) Gives you the diagonal of the square. The diagonal of a square creates two 45/45/90 triangles.

Use this knowledge to find the other side lengths and then the perimeter.