 Hello friends, welcome again to another session on Gems of Geometry Continuing with our trend We are going to discuss another very important result here so the result is that the if you have a triangle ABC and AD is One of the altitudes then the length AD will be equal to B times C divided by 2 R where R is the circum radius of the triangle ABC Okay, so AD as you can see here AD I have shown the value also in this case. It is 5.88 B is the side opposite to angle B B here B. This is angle. This is side C so hence AD will be equal to B times C divided by twice of R R is the circum radius so What we'll do is we'll just change the position of C let's say and you can now see that that whatever is the position of C we can calculate the value of a B right so it's very much possible So here it is so you can you can see if you even if I change the position of B It remains the same correct or a So hardly matters right so hence you'll get the same Value right so AD length of the Hypotenuse or sorry length of the order altitude is equal to Length of the adjacent sides product of the adjacent sides Right. So for example AD is the altitude so in the adjacent sides in this case will be C and B So product of these two sides divided by two times the Circum radius. That's what is the theorem. Let's try to prove this So let's prove this theorem now. So if you can see in triangle ABD so AB and D this triangle ABD and A A dash C. So A A dash C if you see both these triangles are similar. Why? because first of all ADB Angle ADB You can see here ADB is 90 degrees and A C A dash will also be 90 degrees. Why? Because A 0 8 4 A dash happens to be the diameter and diameter subtends an angle 90 degree at any Quantum circle right so this we know so it's both of them are 90 degrees Also ABB or other ABC which are same A again A B D which is equal to A B C this angle will be equal to A A dash C right A A dash C because A C being the same chord So it is subtending two angles at point B and a dash Right. So hence what will happen these two angles are also equal. So if these two angles are equal then what happens the two The two triangles will be similar by a criteria a similarity criteria. So hence their corresponding sites must be proportional So hence AD this AD upon the altitude here AC so AD upon AC will be equal to AB upon A A dash right that's what I have written here right so corresponding parts of Similar triangles so AD is written as it is but AC was B You can see AC is B Similarly AB was C and a a dash clearly diameter so to bar So clearly by cross multiplying you can see AD is BC upon to R, right? This is what we intended to prove that means We got this result that altitude length will be equal to product of the adjacent sides You can see adjacent sides B and C here and divided by twice the circum radius, right? So any altitude Can be expressed as the product of the two sides and the diameter or the twice the circum radius. That's the rule for the theorem