Squares - GMAT Quantitative

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.

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.

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.