 Hello and welcome to the session. In this session we are going to discuss the following question and the question says that, find measure of angle LNP from the given figure. Before starting with solution of this question, we should know our result. Law of cosine states that if there is any triangle A, B, C with A, B and C representing the measure of size opposite to angles with measurements A, B and C respectively, then the following equations are true. That is, A square is equal to B square plus C square minus 2 BC into cos of angle A. B square is equal to A square plus C square minus 2AC into cos of angle B and C square is equal to A square plus B square minus 2AB into cos of angle C. With this key idea, let us proceed to the solution. We are given a figure of rectangular box with height 6 cm, length 13 cm and breadth 5 cm. We are given triangle LNP and we have to find measure of angle LNP. For this we find measure of all the sides of this triangle that is, side LN, let us denote the length of this side as P, side NP, let us denote the length of this side as L and side LP which is denoted by N. First we find the measure of side LN which is denoted by P. Now we consider this triangle LON which is a right angle triangle where the length of the side LO is equal to 5 cm, the length of the side ON will be equal to 13 cm. And we have taken the length of the side LN as P which we have to find. Now using Pythagorean theorem here, we have LN square is equal to LO square plus ON square. Now putting in the values of LO, ON and LN in this equation, we get P square is equal to LO square that is 5 square plus ON square that is 13 square. Which implies that P square is equal to 25 plus 169 which implies that P square is equal to 194. Now taking positive square root on both sides we get P is equal to square root of 194 which implies that P is equal to 13.92. So length of the side LN will be equal to 13.92 cm. Now we consider the triangle PQN which is a right angle triangle where the length of the side PQ is given as 13 cm, the length of the side QN is given as 6 cm and we have taken the length of the side NP as L. So now using Pythagorean theorem here we get NP square is equal to PQ square plus QN square. Now substituting the values of NP, PQ and QN here, we get L square is equal to 13 square plus 6 square which further implies that L square is equal to 169 plus 36 which implies that L square is equal to 205. Now taking positive square root on both sides we get L is equal to square root of 205 which further implies that L is equal to 14.3. So we say that the length of the side NP is equal to 14.3 cm. Now again if we take this right angle triangle that is triangle LSP where length of the side SP is equal to 5 cm, the length of the side LS is equal to 6 cm and we have taken the length of the side LP as N. Using Pythagorean theorem here we have LP square is equal to SP square plus LS square. Now we substitute the values of LP, SP and LS here in this equation and we get N square is equal to 5 square plus 6 square which further implies that N square is equal to 25 plus 36 that is N square is equal to 61. Now taking positive square root on both sides we get N is equal to square root of 61 which implies that N is equal to 7.8. So we say that the length of the side LP is equal to 7.8 cm. Now we have all the three sides of the triangle LNP where P is equal to 13.92 cm, L is equal to 14.3 cm and N is equal to 7.8 cm. Now from the key idea using law of cosines we have N square is equal to L square plus P square minus 2 LP into cos of angle theta or we can write this equation as theta is equal to cos inverse of L square plus P square minus of N square whole upon 2 into L into P. Now we will substitute the values of LPN in this equation and we get theta is equal to cos inverse of N square that is 14.3 square plus P square that is 13.92 square minus of N square that is minus of 7.8 square whole upon 2 into L that is 2 into 14.3 into P that is 13.92. This implies that theta is equal to cos inverse of 204.49 plus 193.7 minus 60.84 whole upon 398.112. On solving further we get the value of theta as cos inverse of 337.35 upon 398.112 which implies that theta is equal to cos inverse of 0.84. Now using calculator we find the value of cos inverse of 0.84 and we get theta is equal to 32.85 degrees thus measure of angle LNP is equal to 32.85 degrees This is the required answer. This completes our session. Hope you enjoyed this session.