Quadrilaterals - GMAT Quantitative

In order to find the diagonal, we must know the sides of the rectangle or know whether the triangles ADC or ABD have special angles.

Statement 1 alone doesn't let us calculate the hypothenuse of the triangles, because we only know one side.

Statement 2 alone is sufficient because it allows us to find all angles of the triangles inside of the rectangle. We can see that they are special triangles with angles 30-60-90. Any triangle with these angles will have its sides in ratio , where is a constant. Here, , knowing this, we can calculate the length of the hypothenuse, also the diagonal, which will be .

Hence, statement 2 is sufficient.

I) Gives us the length of ASOF's diagonal. This by itself does not give us any way of finding the other sides.

II) Gives us one side length. From there we can use the perimeter to find the other side length and then the area.

Therefore, Statement II is sufficient to answer the question.

The length of a diagonal of a rectangle can be calculated like a hypotenuse using the Pythagorean Theorem provided we have information about the lengths of the rectangle.

Statement 1 tells us that both triangles ADC and ABD have their angles in ratio , which means that their sides will have length in ratio , where is a constant. We can't tell however what length the diagonal will be.

Statment 2 tells us that side AC is 1. From there we can't conclude anything. Indeed, rectangle ABCD might as well be a square or a very thin rectangle, we don't know.

Both statements together however, allow us to tell that and therefore that the diagonal will be 2.

Hence, both statements together are sufficient.

To solve this, we need information about the lengths of the sides or to whether triangles ADB and ACD are special triangles.

Statement 1 tells us that CDA has length 3. This is not enough and we still don't know whether the rectangle is of a special type of rectangle.

Statement 2 tells us that triangles ADB and ACD are special triangles, indeed, they have their angles in ratio . That means that their sides will be in ratio . Now we don't need to know what is constant , since it will cancel out in the ratio.

Therefore, statement 2 alone is sufficient.

In order to find the diagonal of a rectangle we need the length of both sides.

We can't find the lengths with just the area or just the perimeter, but by using them both together we can make a small system of equations with two unknowns and two equations.

Then we can solve for each side and use the Pythagorean Theorem to find our diagonal.

In order to find the diagonal, we must know the sides of the rectangle or know whether the triangles ADC or ABD have special angles.

Statement 1 alone doesn't let us calculate the hypothenuse of the triangles, because we only know one side.

Statement 2 alone is sufficient because it allows us to find all angles of the triangles inside of the rectangle. We can see that they are special triangles with angles 30-60-90. Any triangle with these angles will have its sides in ratio , where is a constant. Here, , knowing this, we can calculate the length of the hypothenuse, also the diagonal, which will be .

Hence, statement 2 is sufficient.

I) Gives us the length of ASOF's diagonal. This by itself does not give us any way of finding the other sides.

II) Gives us one side length. From there we can use the perimeter to find the other side length and then the area.

Therefore, Statement II is sufficient to answer the question.

The length of a diagonal of a rectangle can be calculated like a hypotenuse using the Pythagorean Theorem provided we have information about the lengths of the rectangle.

Statement 1 tells us that both triangles ADC and ABD have their angles in ratio , which means that their sides will have length in ratio , where is a constant. We can't tell however what length the diagonal will be.

Statment 2 tells us that side AC is 1. From there we can't conclude anything. Indeed, rectangle ABCD might as well be a square or a very thin rectangle, we don't know.

Both statements together however, allow us to tell that and therefore that the diagonal will be 2.

Hence, both statements together are sufficient.

To solve this, we need information about the lengths of the sides or to whether triangles ADB and ACD are special triangles.

Statement 1 tells us that CDA has length 3. This is not enough and we still don't know whether the rectangle is of a special type of rectangle.

Statement 2 tells us that triangles ADB and ACD are special triangles, indeed, they have their angles in ratio . That means that their sides will be in ratio . Now we don't need to know what is constant , since it will cancel out in the ratio.

Therefore, statement 2 alone is sufficient.

In order to find the diagonal of a rectangle we need the length of both sides.

We can't find the lengths with just the area or just the perimeter, but by using them both together we can make a small system of equations with two unknowns and two equations.

Then we can solve for each side and use the Pythagorean Theorem to find our diagonal.

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.