Rectangles - GMAT Quantitative Reasoning

To find the rectangle, you need the length and the width.

If you know the diagonal and the angle it forms with one of the longer sides, you can use trigonometry to find both length and width:

From there, the area follows.

If you know only the diagonal, you have insufficient information; the length and width can vary according to that angle. If you only know the angle, you can discern the proportions of the sides, but not the actual lengths.

The portion of the rectangle to the right of the -axis has area ; to the left, . From Statement 1 alone, since , , and the portion of the rectangle on the right is greater than the portion on the left. However, this is all we can determine.

By a similar argument, from Statement 2 alone, the portion of the rectangle above is greater than the portion below, but this is all we can determine.

From both statements together, however, we can compare the portions of the rectangles in the four quadrants. The areas of each are:

Quadrant I:

Quadrant II:

Quadrant III:

Quadrant IV:

Since and , is the greatest of the four quantities, and we see that Quadrant I includes the lion's share of the rectangle.

The figure referenced is below:

Assume Statement 1 alone. is a diagonal of the rectangle, and also a diameter of the circle. Statement 1 gives the area of the circle, from which the radius, and, subsequently, , can be calculated. However, infinitely many rectangles of different areas can be constructed in this circle, so without any further information, it is not clear what the sidelengths are - and what the area is.

Assume Statement 2 alone. This statement only gives the length of one side. Without any further information, the area of the rectangle is unknown.

Now assume both statements are true. can be calculated, and is given, so the Pythagorean Theorem can be used to find . The area of the rectangle is the product , so the two statements together are sufficient.

The figure referenced is below (note that the figure itself assumes Statement 2, but this is not known from Statement 1):

Assume Statement 1 alone. A diagonal of a rectangle inscribed inside a circle is a diameter of the circle. Statement 1 gives the area of the circle, from which the radius, and, subsequently, , can be calculated. However, infinitely many rectangles of different areas can be constructed in a given circle, so without any further information, it is not clear what the sidelengths are - and what the area is.

Assume Statement 2 alone. It follows that all of the sides of the rectangle/square are congruent, but without the common sidelength, the area of the square cannot be calculated.

Assume both statements. can be calculated, and the area of the square can be calculated to be .

A a rectangle with sides of length 1 and 4 and a rectangle with sides of length 2 and 3 both have perimeter 10, but they have different areas ( 4 and 6, respectively), making Statement 1 alone inconclusive. Statement 2 is inconclusive, there being infinitely many primes.

Assume both statements.

Then

Since and are both prime integers, one must be 2 and the other must be 3 (1 and 4 cannot be a possibility, since 1 is not a prime). It does not matter which is which, so the numbers can be multiplied to obtain area 6.

Assume both statements. Then from Statement 1, it follows that:

There are two pairs of odd primes that add up to 18 - (5,13), in which case the area is 65, and (7,11), in which case the area is 77. The two statements together are inconclusive.

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.

The length of one diagonal alone does not prove the parallelogram to be a rectangle, nor do the lengths of the sides.

Suppose we know all of these lengths, though. Since is a parallelogram, if , then .

The sides and diagonal form a triangle with sidelengths 25, 60, and 65. The parallelogram is a rectangle if and only if is a right angle; therefore, we must determine whether the conditions of the Pythagorean Theorem hold:

This is true; is a right angle and is a rectangle.

Therefore, both statements together are sufficient to answer the question, but neither statement alone is sufficient to answer the question.

To find the area we need the length and width of the rectangle. We can use II together with I to make an equation for perimeter with only one unknown.

So we need both to solve.

Solve for and then go back to find and then with that you can find the area of the base and you are finished.

When asked to find the width of a rectangle we will need to use both statemests together.

For Statement I) we can use the perimeter formula.

Now, for Statement II) we will use the length of the diagonal along with the Pythagorean Theorem.

From here you can solve the perimeter equation in terms of either l or w. Then you can use substitution into the Pythagorean Theorem to solve for a possible width.

Statement 1:

Recall the formula to find the perimeter of a rectangle. Substitute in the given information and solve.

Statement 2:

Recall the formula for the area of a rectangle. Substitute in the given information and solve.

Each statement alone is sufficient to answer the question.

Statement 1:

Statement 1 is sufficient to answer the question

Statement 2:

Statement 2 is also sufficient to answer the question

To know the perimeter of a rectangle, we need to know length and width.

Statement (1) provides Area, which does not provide the necessary information. NOT SUFFICIENT

Statement(2) provides one side, but we still need the other to determine perimeter. NOT SUFFICIENT.

Both together are sufficent however since if we know one side and the area, we can find the remaining side using division, and then use the two side lengths to find the perimeter.

Let be the sidelength of the square. Then its perimeter is .

From Statement 1 alone, the length of the rectangle is , and, if is the width, the perimeter of the rectangle is . Therefore, we can prove that the rectangle has the greater perimeter.

From Statement 2 alone, the width of the rectangle is , and its perimeter is at least - but unless we know the length, we do not know whether the total perimeter is greater than, equal to, or less than .

To find perimeter we need length and width. We are given the length.

I) Is irrelevant.

II) We are given a way of relating area and width. Since we know that area is length times width, we can use II to set up an equation where we substitute in the known length along with the given statement to solve for our width

Solve the second one for w and you're good to go!

.

Therefore the perimeter would be: