Hello and welcome to the session. My name is Mansi and I'm going to help you with the following question. The question says two cords A, B and C, D of lengths 5 cm and 11 cm respectively of a circle are parallel to each other and are on opposite sides of its center. If the distance between A, B and C, D is 6 cm, find the radius of the circle. Now we see in the question that we are given a circle with center O and two cords A, B and C, D on the opposite sides of the center such that A, B is parallel to C, D and the distance between A, B and C, D is 6 cm. Now in this question we have to find the radius of the circle. So let us start with the solution to this question. So let O be the center of the given circle and let radius be r cm. Now we draw O, P perpendicular to the cord A, B and O, Q perpendicular to the cord C, D. So now we draw O, P perpendicular to A, B and O, Q perpendicular to C, D. We have A, B is parallel to C, D because that is given to us in the question. Therefore points A, O and Q are collinear. So P, Q is equal to 6 cm. Now what we do? We let O, P to be equal to x. So O, Q becomes 6 minus x cm because this entire distance that is P, Q is 6 cm. So if we have taken O, P as x cm then O, Q becomes 6 minus x cm. Now we do some construction in this figure. First of all we join O to A and O to C. Now we see that O, A is equal to O, C is equal to r that means this is the radius of this circle. Now since the perpendicular from the center to a cord bisets the cord since the perpendicular from the center to a cord of the circle bisets the cord. Therefore we have P is equal to 5 by 2 cm and C, Q is 11 by 2 cm. So we see that A, O is our cm, this is x cm and A, P is 5 by 2 cm and P, Q is equal to 11 by 2 cm. Now we see that triangle O, A, P and triangle O, C, Q this is these are two right angle triangles because angle O, P, A is equal to angle O, Q, C and both are 90 degree each. So these are right angle triangles. So we can say that in right triangles O, P, A and O, Q, C we will have r square is equal to 5 by 2 the whole square plus x square and also we will have r square is equal to 6 minus x the whole square plus 11 by 2 the whole square. So in these two triangles we have r square is equal to 5 by 2 the whole square plus x square this we get by hypotenuse theorem that says that the square of hypotenuse is equal to the square of the base plus square of the height. So we have this and r square is also equal to 11 by 2 the whole square plus 6 minus x the whole square. Now we see in the figure that we took O, P to be equal to x cm so O, Q became 6 minus x cm but if we would have taken O, Q to be equal to x cm then O, P becomes 6 minus x cm so we can say this happens or we could also say that r square is equal to 5 by 2 the whole square plus 6 minus x the whole square and r square is equal to 11 by 2 the whole square plus x square. Now we see that when solving these two equations we can easily find out the value of x so we see that this is equal to r square and this is also equal to r square so this is equal to this. We have 11 by 2 the whole square plus x square is equal to 5 by 2 the whole square plus 6 minus x the whole square. Now this implies that 11 by 2 the whole square minus 5 by 2 the whole square is equal to 6 minus x the whole square minus x square. Now let us open the brackets on both the sides. We see that 11 square is 121 by 2 square is 4 minus 5 square is 25 and 2 square is 4. Now this is equal to here we use the formula A minus B the whole square. We see that 6 minus x the whole square becomes 36 that is 6 square. Now plus x square minus 2 into 6 into x that is 12x and minus x square this comes here as it is. Now we see that plus x square gets cancelled with minus x square. Now we see that on the left hand side we have in both the terms denominator is same so we can write it as 121 minus 25 by 4 is equal to 36 minus 12x. Now 121 minus 25 is 96 by 4 is equal to 36 minus 12x. Now 96 divided by 4 is we divide the numerator denominator by 4 and the denominator will have 1 and the numerator will have now we see that 4 2s are 8 we have 1 left over 1 goes there we know that 2 8s are 16 we know that 2 4s are 16 so this becomes 24 in the numerator we have 24 minus 36 is equal to minus 12x. We see that 24 minus 36 is minus 12 and this is equal to minus 12x. On dividing both the sides by minus 12 we have x is equal to 1. Now since we are asked to find out the radius in this question so now we can put this x equal to 1 in any one of these two equations. So we put it here we see that 11 by 2 the whole square plus x square is equal to r square now we put x equal to 1 we get 11 by 2 the whole square plus 1 square is equal to r square this implies 121 by 4 plus 1 is equal to r square. Now let us add these two terms in the second term we have the denominator 1 so we take the LCMO 4 and 1 that is 4 in the numerator we have 121 plus 4 is equal to r square 121 plus 4 is 125 denominator remains as it is we have r square. Now since we have to find the value of r and not r square so we take square root on both the sides we have r is equal to under the root 125 by 4. Now this will be equal to plus minus 5 root 5 by 2 centimeter since we have to find out the radius of the circle. So radius of the circle can never be negative so we neglect the negative term and we have 5 root 5 by 2 centimeter which is also our answer to this question. Now let us just summarize what we have seen first of all we take a circle with center O then we draw perpendicular from O to CD and AB we put OQ to be equal to x if OQ is x then OP becomes 6 minus x now we see that these two becomes right angle triangle so we can always use hypotenuse theorem here so then we get two equations of these two triangles one equation is r square equals to 5 by 2 the whole square plus 6 minus x the whole square and other is r square is equal to 11 by 2 the whole square plus x square then we solve these two equations to get the value of x we get the value of x as x is equal to 1 here we again see that x is the length of perpendicular from O on the chord of length 11 centimeter now then we put x equal to 1 in this equation to get the value of r and finally we get the value of r as 5 root 5 by 2 centimeter and this becomes our answer to the question I hope you understood the question and you enjoyed the session have a good day.


The Japanese fighters joining Ukraine's fight. Japanese men have begun to join Ukrainians battling the Russian invasion, defying their government's warnings and booking a decade-long national principle of passivism. Japan's military is constitutionally limited to defence and has not fought since World War II. Georgian Legion commander Mamuka Mamoulashvili told AFP the unit currently counts 8 Japanese among troops from 33 nationalists in its ranks. They are very motivated, very disciplined and easily master the training they are undergoing now, he said. While many of the foreign fighters who have flocked to Ukraine from other countries come with military and combat experience, Japan's unique constitutional constraints mean its volunteers start as absolute novices. Japan's government, like those elsewhere, has warned its citizens against travel to Ukraine. A small, determined group has ignored that advice. Last year, Japan confirmed a citizen who was reportedly a former member of the country's military had been killed in combat in Ukraine. People in Ukraine may feel encouraged just by the fact that we came here all the way from Japan. Yuya Motomura, a Japanese man who had joined Ukrainians battling the Russian invasion, said AFP from Kyiv. I am here in this country hoping to make use of myself. In that sense, I already feel fulfilled, he added.