Quadrilaterals - GRE Quantitative Reasoning

All sides are equal in a square. To find the area of a square, multiply length times width. We know length = 4 but since all sides are equal, the width is also 4. 4 * 4 = 16.

A diagonal divides a square into two 45-45-90 triangles, which have lengths adhering to the ratio of x: x: x√2. Therefore, 2 is the correct answer as the diagonal represents the hypotenuse of the triangle. the Pythagorean theorem can also be used: √22+ √22 = c2.

All sides are equal in a square. To find the area of a square, multiply length times width. We know length = 4 but since all sides are equal, the width is also 4. 4 * 4 = 16.

A diagonal divides a square into two 45-45-90 triangles, which have lengths adhering to the ratio of x: x: x√2. Therefore, 2 is the correct answer as the diagonal represents the hypotenuse of the triangle. the Pythagorean theorem can also be used: √22+ √22 = c2.

There are two components to solving this geometry puzzle. First, one must be aware that the sum of the measures of the interior angles of a parallelogram is 360 degrees (sum of the interior angles of a figure = 180(n-2), where n is the number of sides of the figure). Second, one must know that the other two interior angles are doubles of those given here. Thus if we assign one interior angle as x and the other as x-25, we find that x + x + (x-25) + (x-25) = 360. Combining like terms leads to the equation 4x-50=360. Solving for x we find that x = 410/4, 102.5, or approximately 103 degrees. Since x is the measure of the larger angle, this is our answer.

Because of the character of parallelograms, we know that our figure can be redrawn as follows:

Because it is a four-sided figure, we know that the sum of the angles must be . Thus, we know:

Solving for , we get:

By using the properties of parallelograms along with those of supplementary angles, we can rewrite our figure as follows:

Recall, for example, that angle is equal to:

, hence

Now, you know that these angles can all be added up to . You should also know that

Therefore, you can write:

Simplifying, you get:

Now, this means that:

and . Thus, the two values are equal.

There are two components to solving this geometry puzzle. First, one must be aware that the sum of the measures of the interior angles of a parallelogram is 360 degrees (sum of the interior angles of a figure = 180(n-2), where n is the number of sides of the figure). Second, one must know that the other two interior angles are doubles of those given here. Thus if we assign one interior angle as x and the other as x-25, we find that x + x + (x-25) + (x-25) = 360. Combining like terms leads to the equation 4x-50=360. Solving for x we find that x = 410/4, 102.5, or approximately 103 degrees. Since x is the measure of the larger angle, this is our answer.

Because of the character of parallelograms, we know that our figure can be redrawn as follows:

Because it is a four-sided figure, we know that the sum of the angles must be . Thus, we know:

Solving for , we get:

By using the properties of parallelograms along with those of supplementary angles, we can rewrite our figure as follows:

Recall, for example, that angle is equal to:

, hence

Now, you know that these angles can all be added up to . You should also know that

Therefore, you can write:

Simplifying, you get:

Now, this means that:

and . Thus, the two values are equal.

There are two components to solving this geometry puzzle. First, one must be aware that the sum of the measures of the interior angles of a parallelogram is 360 degrees (sum of the interior angles of a figure = 180(n-2), where n is the number of sides of the figure). Second, one must know that the other two interior angles are doubles of those given here. Thus if we assign one interior angle as x and the other as x-25, we find that x + x + (x-25) + (x-25) = 360. Combining like terms leads to the equation 4x-50=360. Solving for x we find that x = 410/4, 102.5, or approximately 103 degrees. Since x is the measure of the larger angle, this is our answer.

Because of the character of parallelograms, we know that our figure can be redrawn as follows:

Because it is a four-sided figure, we know that the sum of the angles must be . Thus, we know:

Solving for , we get:

By using the properties of parallelograms along with those of supplementary angles, we can rewrite our figure as follows:

Recall, for example, that angle is equal to:

, hence

Now, you know that these angles can all be added up to . You should also know that

Therefore, you can write:

Simplifying, you get:

Now, this means that:

and . Thus, the two values are equal.

One potentially helpful first step is to draw the rectangle described in the problem statement:

After that, it's a matter of using the other information given. The perimeter is given as 14, and can be written in terms of the length and width of the rectangle:

Furthermore, notice that the diagonal forms the hypotenuse of a right triangle. The Pythagorean Theorem may be applied:

This provides two equations and two unknowns. Redefining the first equation to isolate gives:

Plugging this into the second equation in turn gives:

Which can be reduced to:

or

Note that there are two possibile values for ; 3 or 4. The one chosen is irrelevant. Choosing a value 3, it is possible to then find a value for :

This in turn allows for the definition of the rectangle's area:

So Quantity B is 12, which is less than Quantity A.

Based on the information given, we know that could be either the longer or the shorter side of the similar rectangle. Similar rectangles have proportional sides. We might need to test both, but let us begin with the easier proportion, namely:

as

For this proportion, you really do not even need fractions. You know that must be .

This means that the figure would have a perimeter of

Luckily, this is one of the answers!

For this problem, you need to find the pair of sides that would reduce to the same ratio as the original set of sides. This is a little tricky at first, but consider the set:

and

For this, you have:

Now, if you factor out , you have:

Thus, the proportions are the same, meaning that the two rectangles would be similar.

One potentially helpful first step is to draw the rectangle described in the problem statement:

After that, it's a matter of using the other information given. The perimeter is given as 14, and can be written in terms of the length and width of the rectangle:

Furthermore, notice that the diagonal forms the hypotenuse of a right triangle. The Pythagorean Theorem may be applied:

This provides two equations and two unknowns. Redefining the first equation to isolate gives:

Plugging this into the second equation in turn gives:

Which can be reduced to:

or

Note that there are two possibile values for ; 3 or 4. The one chosen is irrelevant. Choosing a value 3, it is possible to then find a value for :

This in turn allows for the definition of the rectangle's area:

So Quantity B is 12, which is less than Quantity A.

Based on the information given, we know that could be either the longer or the shorter side of the similar rectangle. Similar rectangles have proportional sides. We might need to test both, but let us begin with the easier proportion, namely:

as

For this proportion, you really do not even need fractions. You know that must be .

This means that the figure would have a perimeter of

Luckily, this is one of the answers!

For this problem, you need to find the pair of sides that would reduce to the same ratio as the original set of sides. This is a little tricky at first, but consider the set:

and

For this, you have:

Now, if you factor out , you have:

Thus, the proportions are the same, meaning that the two rectangles would be similar.