Toph - Problem Nine and Three Quarter [ Tutorial + Solution ]

=>Try Yourself First. For help Scroll Down .

Today I solved a nice geometry problem. After solving this problem I thought it would be better to write a tutorial. Anyway, let's start.

Problem Link: https://toph.co/p/problem-number-nine-and-three-quarter

Problem Clearance: 

The problem said to find the value of HI where the radius of the arc is given and the value of <ABC angle is given. It's been mentioned that D is the midpoint of BC and angle <ABN is half of angle <ABC.

Additional Drawing: 

Let's first denote the intersection point of HI and BN is O . Now let's connect AC & AN . Now draw another perpendicular line from D which intersect HB at E and BN at F . The value of angle <ANB = <ACB = 90 degrees. Because if we draw two line from two end point of a chord that connects to the same point on the arc then we can be so sure that it will be a right angle.

Procedure: 

We know how to find out the two side of a triangle if we have the value of a single side and two angles. That is the sine rule -

a / sin A = b / sinB = c / sinC 

Now, we have the angle <ABC and angle <ACB and AB = 2 * HB . So we can easily find out the value of angle <BAC and length of AC & BC . As D is the mid point of BC then we also have BD . From this, we can find out angle <BDE and the side length of DE & EB. Now for triangle BEF we can figure out angle <BFE and the side length of FE & BF. 

And from triangle ABN we have angle <ABN , angle <ANB and side length AB. So it's easy to find out angle <BAN and side length BN & AN. 

We know, for a right triangle,  tanX = Height / Base ( where X is the angle between Hypotenous & Base ). We have HB and angle <HBO . Hence, we can find out HO. So now we have to figure out IO . 

Now to figure out IO, we consider the triangle NFD and triangle NOI . Both are similar triangles. A similar triangle has an angle of the same value. In that case, the similar angle is <FND = <OND . So, for a similar triangle, we can say, 

IO / DF = NO / NF 

So DF = DE - EF , NF = NB - BF , NO = NB - OB .

To find out OB , we can use Pythagoras rule, OB^2 = OH^2 + HB^2 . 

So now we have all the values. After finding out IO just add it with HO . Hence we have the result. 

=============================Hope you solved it by now ============================= 

Source Code: GitHub Link 

=>Leave a comment for any kinds of Hugs and / or Bugs. Thank you .