 Hello and welcome to the session. In this session we will discuss the statement and proof of Pythagoras theorem and its converse. Now Pythagoras theorem holds for right angle triangles. Now a right angle triangle is a triangle which has a right angle that is an angle measuring 90 degrees as one of its angles. Now suppose A B C is the right angle triangle, right angle at B. Now here the side opposite to the right angle is called hot puppy news. This is the longer side of the triangle and the other two sides are called less of the triangle. Now when we say a right angle triangle A B C right angle at B it means the right angle is found at the vertex B. So here angle B is 90 degrees, A C is the hot puppy news and A B and B C are the legs of triangle A B C. Now let us discuss this Pythagoras theorem. Now Pythagoras theorem states that in a right angle triangle sum of squares of length of legs is equal to square of length of hot puppy news. Now if in a right angle triangle length of hot puppy news is C and length of legs are A and V then according to Pythagoras theorem we have one of length of hot puppy news that is C square is equal to sum of squares of length of legs that is A square plus B square. Now see the following right angle triangles. Now in figure one length of hot puppy news is C and length of other two sides of this triangle are A and B. Now when this triangle is rotated at an angle of 90 degrees we get figure two and again when this triangle in figure two is rotated at an angle of 90 degrees then we get figure three and lastly when this triangle that is the triangle in figure three is rotated at an angle of 90 degrees we get figure four. Now here we can see that all the four triangles formed are identical or you can say all these triangles are congruent because all corresponding sides and angles of these triangles are equal. Now when we draw angles together get this figure which forms a square. Now here two squares are formed the bigger square is formed by the legs of these triangles. So here its side is of length A plus B. So here length of each side of this bigger square is A plus B and this smaller square is formed in the center by the high continuous of length C. So each side of inner square is of length C. And now we will start with the proof of Parker's theorem. Now here on a square we draw four congruent right angle triangles that is triangle EED triangle EQF triangles FGR and DSG. Then a smaller square DEFG is formed in the center. Now from the figure we can see length of each side of larger square is equal to sum of length of legs of triangle which is equal to A plus B and length of side of small square is equal to length of height of news which is equal to C. Now this larger square is formed by these four triangles and the smaller square so total area of larger square is equal to area of four triangles smaller square. Now we know that area of a square is equal to square of its side. So this implies area of larger square will be A plus B whole square as length of each side of larger square is A plus B and this is equal to area of four triangles. Now area of each triangle will be equal to half into base into height. So area of four triangles will be four into one by two into A into B the whole plus area of smaller square will be smaller square is and this implies now A plus B whole square is A square plus B square plus two AG and this is equal to now two into two is four so this will be two into A into B which is two AG plus C square further this implies A square plus B square plus two AG minus two AG is equal to C square now plus two AG will be cancelled with minus two AG and this implies A square plus B square is equal to C square. So here we have proved that in a right angle triangle sum of squares of length of legs is equal to square of length of fiber news. Hence we have proved the Pythagoras theorem. Now let us see the use of this theorem. Now let us discuss an example for this. Now in a right angle triangle when we know length of any two sides when using Pythagoras theorem we can find length of the third side. Now here in this right angle triangle we have to find the value of X. Now we know that in a right angle triangle side opposite to the right angle is called the hypotenuse. So here one meter represents hypotenuse and length of one of its legs is given as root three by two meters and we have to find the length of other leg. Now by Pythagoras theorem we know that in a right angle triangle square of the hypotenuse that is square of the length of hypotenuse which will be one square is equal to sum of the squares of length of its legs which is root three by two whole square plus X square. Now this implies one is equal to three by four plus X which implies X square is equal to one minus three upon four which is equal to four minus three whole upon four which is equal to one upon four and this implies X is equal to square root of one upon four. Now taking positive square root because length cannot be negative. So here X will be equal to one upon two. So here length of this leg is one upon two meters. Now let us discuss converse of Pythagoras theorem. Converse of Pythagoras theorem states that if triangle has sides of length a b and c and a square plus b square is equal to c square then this triangle is a right angle triangle. Now let us start with its roots. Now here it is given triangle of length a b and c that is pq is equal to a qr is equal to b and pr is equal to c and also it is given that a square plus b square is equal to c square and you can write pq square plus qr square is equal to pr square and we have to prove that triangle pqr is a right angle triangle. Now for the proof of converse of Pythagoras theorem we shall make use of congruence of two triangles. Now we know that in congruent triangles corresponding parts are equal that is corresponding sides and corresponding angles are equal. Now for the proof we will construct another triangle a b c such that angle b is equal to 90 degrees side a b is equal to pq is equal to a side b c is equal to qr is equal to b. Now let us start with its roots a b c is a right angle triangle so in triangle a b c Pythagoras theorem a c square is equal to a b square plus b c square here side a b is equal to pq and side b c is equal to qr so this implies a c square is equal to pq square plus qr square now also it is given that pq square plus qr square is equal to pr square so this implies a c square is equal to now taking positive square root this implies a c is equal to pr. Now in triangle pqr and triangle a b c pq is equal to a b qr is equal to b c pr is equal to right side side side property triangle pqr is congruent to triangle a b c. Now we know that corresponding parts of congruent triangles are equal so here angle q will be equal to angle b which is equal to 90 degrees so here we have proved that angle q is equal to 90 degrees hence we have proved that triangle pqr is a right angle triangle right angle at q. Now to see the use of converse now suppose we are given any three numbers say three four and five and we have to find whether they form a right angle triangle or not then we make use of converse of Pythagore theorem then we check that three square plus four square is equal to five square or not now here you have to take the square of the greatest number on one side and the sum of the squares of other two numbers on the other side now five square is equal to 25 and three square plus four square is equal to 9 plus 16 which is also 25 so three square plus four square is equal to five square so according to the converse of Pythagore theorem they form sides of right angle triangle with high party news such numbers are called Pythagorean triplet so in this session you have learnt the statement and proof of Pythagore theorem and its converse and this completes our session hope you all have enjoyed the session.