NotThatSmart

We can set up variables to complete this problem. 

First, le'ts graph this problem. We have

Now, let's set up our first variable. 

Let's let the side of square be x. Since the base of the triangle is the same length as the side of the square, we have

base of triangle = x as well. 

Next, from the graph, which we nicely split into quarters, note that

height of triangle = \(\frac{3x}{4}\)

Now, since we have the base and height of the triangle, we can find it's area in terms of x. We have

\(\frac{3x}{4}\cdot x \cdot \frac{1}{2} = \frac{3x^2}{8}\)

The area of the square is just the side squared, so 

Area of square = \(x^2\)

Thus, we have what we need. 

\(ratio = \frac{3x^2}{8} \cdot \frac{1}{x^2} = \frac{3}{8}\)

So our answer is just 3/8. 

Thanks! :)

The rectangle with these coordinates is centered at the origin, with side length 2. Note that any equation y = mx will bisect the area of the square, meaning that essentially the line y=mx will actually go through the origin of the entire graph. 

As seen in the image below. 

Thus, the y intercept is essentially just 0.

Thanks! :)

We essentially have a system of equations to solve for x and y. The equation of

\( x^2 + y^2 = 1\\ y = x\)

Now, from the second equation, we know that x=y. Thus, subsituting out y for x, we have

\(x^2+x^2=1\\ 2x^2=1\\ x^2=1/2\\ x=\pm \frac{1}{\sqrt2} = \pm\frac{\sqrt2}{2}\)

Since x is equal to y, we have two points. We have

\((\frac{\sqrt2}{2},\frac{\sqrt2}{2}), (-\frac{\sqrt2}{2},-\frac{\sqrt2}{2})\)

Using the distance formula on these two points (or you could just graph it and count), we get

\(\sqrt{\left(-\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}\right)^2+\left(-\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}\right)^2}=2\)

So 2 is the answer. 

Thanks! :)

If angle XWZ = 60°, angle WXY = 180° - 60° = 120° because XY and WZ are parallels.

Angle ZXW must be less than 120°.

This problem is therefore invalid. 

Thanks! :)

In order to solve this problem, we can use a very simple trick. 

First, let's set \(x = 0.\overline{2123}\)

If we have this for the value of x, we get \(10000 x = 2123.\overline{2123}\)

Now, we subtract x from 10000x. Note that the repeating parts get canceled out. We get

\(10000x-x= 2123.\overline{2123} -\overline{2123}\\ 9999x=2123\)

This means we get

\(x = \frac{2123}{9999}\)

Simplifying, we get

\(x = \frac{193}{909}\)

So our final answer is \( \frac{193}{909}\)

Thanks! :)

Let's make some observations. 

Starting with \(2!\), every other term after that is even, since it contains 2. 

Since we know that \(even + even = even\), then we have \(2! + 3! + \dots + 100! = even\)

Thus, the sequence \(2! + 3! + \dots + 100! \equiv 0 (\mod 2)\)

However, we forgot about 1 term. Since , and 1 is odd, and we know , then we have

\(1! + 2! + 3! + \dots + 100! \equiv 1(\mod 2)\)

Thus, the answer is 1. 

Thanks! :)

First, let's put 0.11012 as a fraction. We have

\(\frac{0.11012}{1}\times \frac{100000}{100000}= \frac{11012}{100000}\)

Now, we must find a common factor between the two. Well, 4 is one, so dividing both the top and bottom by 4, we get

\(\frac{11012 \div 4}{100000 \div 4}= \frac{2753}{25000}\)

Unforutnately,  there's no more factors after this, so our final answer is 

\(0.11012 = \frac{2753}{25000}\)

Thanks! :)

mmm...take a look at the points given. 

\(P_1 P_2 + P_2 P_3 + P_3 P_4 + \dots + P_9 P_{10} + P_{10} P_1\)

This is essentially just the perimeter of the decagon \(P_1P_2P_3P_4P_5...P_{10}\)

The lenngth of one side is essentially

\( radius/2 ( -1 +\sqrt 5)\)

Now that we know one side, we can find the perimeter through the equation

\( 10 (1/2) ( -1 + \sqrt 5) = 5 ( -1 + \sqrt 5) ≈ 6.18\)

Thus, the answer is 6.18, 

Thanks! :)