A diagonal divides a square into two 45-45-90 triangles, which have lengths adhering to the ratio of x: x: x√2. Therefore, 4√2 is the correct answer as the diagonal represents the hypotenuse of the triangle. the Pythagorean theorem can also be used: 4²+ 4² = c^2. 

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 = c^2. 

The diagonal separates the square into two 45-45-90 right triangles. The problem can be solved by using the Pythagorean Theorem, a² + b² = c².  It can also be solved by recognizing the 45-45-90 special triangles, which have side ratios of x : x : x√2. 

We know that each side of the square is 6, so use the Pythagorean Theorem to solve for the diagonal of the square. The diagonal of the square is also the diameter of the circle.

Therefore the radius must be 

Now let's find the area inside the circle using the radius.

 meters² 

Quantity A: The diagonal of a square with a side-length of 7.

Quantity B: The side-length of a square with a diagonal of 14.  

Both quantities can be determined, so can the relationship. 

Quantity A:

To determine the diagonal of a square, it's important to remember that a diagonal directly bisects a square from corner to corner. In other words, it bisects the corners, creating two triangles with 45:45:90 proportions, with the diagonal serving as the hypotenuses . If you remember your special triangles, then the side-side-hypotenuse measurements have a ratio of . 

In Quantity A, the side-length is 7. Following the proportion:

The diagonal equals .

Quantity B:

We can use the same ratio to figure out quantity B by substituting x for the unknown side-length quantity, which looks like this:

To find x in this ratio, just isolate x in the hypotenuse:

Divide by 

Now, how does Quantity A and Quantity B match up? 

On the surface, it looks like the two quanties are equal. But how do we prove it?  Well, we know that .  Therefore, we know that:

Divide both sides by 

Therefore, both quantities are equal. 

The first step to a quantitative comparison is to determine whether it can be solved at all with the given knowledge. Since all you need to find the side-length of a square is the perimeter, the area, OR the diagonal and we have one of each for these two quantities, this relationship can be determined. Thus, "the relationship cannot be determined" is out. 

Now, to solve both quantities.  Quantity A can be solved by translating the perimeter into side lengths: the formula for the perimeter of a square is , with  being the side-length, so you just need to divide the perimeter by four.

Thus, quantity A is .

Quantity B can be solved by translating the area into side lengths: the formula for the area of a square is , or , with  being the side-length, so you just need to find the square root of the area.  

Thus, quantity B is roughly .  

Therefore quantity A is greater.  

If the area of Circle  is , we know that the area can be computed using the standard area formula:

, using  for 

Simplifying, we get:

We know that  must be less than . By choosing  for , we find out that this is the radius of our circle. Thus, we know that the diameter of the circle is double this, or . Now, consider the following diagram:

Notice that the diameter is the same length as a side of the square. Thus,  is equal to .

You can further fill in your diagram as follows:

Now, we know that the triangle  is a  triangle. We also know that the length of  and  must also be equal to the diameter of the circle. (The diameter of the circle will run across the circle horizontally if you draw it that way. This will provide you with a complete side length.) Now, we know that the ratio of the hypotenuse of  to the side must be the same as:

For our data, that means:

Simplifying, we know:

Now, make both sides reciprocals:

Finally, solve:

Recall, this is both the length of the side and the diameter of the circle. Hence, you have your answer.

(4,2) and (15,-4)

All that we really need to ascertain is the slope of our line.  So long as a given answer has this slope, it will not matter what its y-intercept is (given the openness of our question).  To find the slope, use the formula: m = rise / run = (y1 - y2) / (x1 - x2):

(2 - (-4)) / (4 - 15) = (2 + 4) / -11 = -6/11

Given this slope, our answer is: y = -6/11x + 57.4

By using the complementary and supplementary rules of geometry (due to lines m and n being parallel), as well as the fact that the sum of all angles within a triangle is 180, we can carry through the operations through stepwise subtraction of 180. 

x = 125 → angle directly below also = 125. Since a line is 180 degrees, 180 – 125 = 55. Since right triangle, 90 + 55 = 145 → rightmost angle of triangle 180 – 145 = 35 which is equal to the reflected angle. Use supplementary rule again for 180 – 35 = 145 = y.

Once can also recognize that both a straight line and triangle must sum up to 180 degrees to skip the last step.