NotThatSmart

Let's note some really important things about this problem. 

If \(AD = AB\), then we have \(\angle ABD = \angle ADB\)

From this, we can write the equation

\(\angle ABD = (180 - 17) / 2 = 81.5° \)

So our answer is just 81.5

Thanks! :)

Unfortunately, I don't think it's used to calculate the angle bisector of a line. 

It is mainly used to calculate if two lines are parrallel to eachother. 

Since \(\frac{Ax+By+C}{\sqrt{A^2+B^2}} \) calculates the distance from the origin for the line \(Ax + By + C =0\) and the same applies for the left hand side, if the two are equal, they are parrallel lines. 

However, I do believe I know what formula was used. 

If we set the slopes of the two lines given in the question to \(m_1\) and \(m_2\), the slope of the angle bisector is

\(m=\sqrt{\frac{m_1m_2+1}{m_1+m_2}}\)

I hope this helps! If you have anymore questions, feel free to ask!

Thanks! :)

We can solve the system of equations for x and y and then apply the distance formula to it. 

We have the system

\(x=4\\ x^2+y^2=25\)

Subsituting x out from the second equation, we have

\(16+y^2=25\\ y^2=9\\ y= \pm 3\)

So the two points where they intersect are at \((4, 3) \text{ and } (4, -3)\)

Now, we simply apply the distance formula to find the two points. We have

\(\sqrt{(4-4)^2+(3-(-3))^2}\\ \sqrt{0+9}\\ \sqrt{9} = 3 \)

So our final answer is 3. 

Thanks! :)

There is a formula similar to the one you showed. 

The equation states that we have \(\frac{Ax+By+C}{\sqrt{A^2+B^2}} = \pm \frac{ax+by+c}{\sqrt{a^2+b^2}}\) for two lines in the form of \(Ax + By + C =0, ax + by + c =0\)

It's similar, but the two lines given are in standard form rather than slope-intercept form.

I'm not too familiar on this equation, but I hope this clarified this a bit.

Feel free to ask if you're still confused. 

Thanks! :)

Let's list out all the cases we have. 

\(1-11\\ 2-10\\ 3-9\\ 4-8\\ 5-7\\ 6-6\\ 7-5\\ 8-4\\ 9-3\\ 10-2\\ 11-1\)

We have 11 ways we can do this. 

Thanks! :)

We can essentially list everything out. 

We have

\(12-0\\ 11-1\\ 10-2\\ 9-3\\ 8-4\\ 7-5\\ 6-6\\ 5-7\\ 4-8\\ 3-9\\ 2-10\\ 1-11\\ 0-12\\\)

As you can see, there are 13 ways we can do this. 

Thanks! :)

Let's note something really important.  

We need to find what \(\angle EPF\) is. 

However, let's read through the problem. It never actually identifies what \(P\)is!

This means it's impossible to solve the problem as P is not defined. 

Thanks! :)

~NTS

Let's note something really important first. 

\(P_1 P_2 + P_2 P_3 + P_3 P_4 + \dots + P_9 P_{10} + P_{10} P_1\) is simply the perimeter of a regular decagon. 

We can find one side of the decagon using the equation \( radius/2 ( -1 + \sqrt{5})\)

So the perimeter is just \(10 (1/2) ( -1 +\sqrt{ 5}) = 5 ( -1 + \sqrt{ 5}) ≈ 6.18\)

So our answer is about 6.18. 

Thanks! :)

~NTS

Before we solve the problem, let's note something important. 

The sum of the angles in a triangle must add up to \(180 ^\circ\)

However, let's take a look at something. 

Adding up the two angles we have, we get \(144 + 66 = 210°\)

This means that triangle AHB would have angles adding up to more than 180. 

Thus, this is an invalid triangle. 

Thanks! :)

~NTS

We can use trial and error to solve this problem. 

First, let's note that

\(582_{10} = N7_{25}\)

Let's decrease to base 20 and see what happens. 

\(582_{10}=192_{20}\)

So the number has to be between 20 and 25. 

Let's try 23. 

We have

\(582_{10}=127_{23}\)

When we convert to 24, we get

\(582_{10}=106_{24}\)

So the smallest number B must be 25. 

So our answer is 25. 

Thanks! :)