 Hello and welcome to the session. The given question says, prove that the ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides and then using the above do the following. The areas of two similar triangles ABC and PQR in the ratio 9 is to 16 and we are given that BC is equal to 4.5 cm. We have to find the length of QR. So, first let us prove the theorem. So, we have two triangles. Let us consider ABC and PQR. We have to prove that area of triangle ABC divided by area of triangle PQR is equal to AB square divided by PQ square is equal to BC square divided by QR square is equal to CA square divided by RB square. That is we have to prove that the ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides. And the construction to this problem is AD perpendicular on BC and PS perpendicular on QR. Now, let us start with the proof. Now, area of triangle ABC divided by area of triangle PQR is equal to now the area of triangle is half of base into height. So, area of triangle ABC is half of BC into AD and therefore area of triangle PQR at the same formula that area of a triangle is half of base into height we can write here that half into PS into QR. Half cancels out the half and we have area of triangle ABC divided by area of triangle PQR equal to BC divided by QR into AD divided by PS. Let this be equation number one. Also, we are given that triangle ABC is similar to triangle PQR. Now, in triangle BT and triangle PQS we will try to show that these two triangles are similar. So, in these two triangles angle ADB is equal to angle Q since we are given that triangle ABC is similar to triangle PQR. Also angle ADB is equal to angle PSQ since each angle is of measure 90 degree. Therefore, angle angle similarity triangle ABD is similar to triangle PQS. Now, this implies that AD divided by PS is equal to AB divided by PQ also since ABC is similar to triangle PQR. Therefore, we have AB divided by PQ equals to BC divided by QR. So, from these two equations we get that AD divided by PS is equal to BC divided by QR. Let this be equation number two. Now, in equation one we have area of triangle ABC divided by area of triangle PQR is equal to BC divided by QR and AD divided by PS. So, in place of AD divided by PS we shall write BC divided by QR. Therefore, we have area of triangle ABC divided by area of triangle PQR equal to BC divided by QR into AD divided by PS we have in place of this we shall write BC divided by QR. So, this implies that area of triangle ABC divided by area of triangle PQR equals to BC divided by QR square. And now, as triangle ABC is similar to triangle PQR we have AB divided by PQ equal to BC divided by QR is equal to AC divided by PR. So, this can further be written as area of triangle ABC divided by area of triangle PQR is equal to BC square divided by QR square and BC divided by QR is equal to AC divided by PR. So, square in both the sides we have AC square divided by PR square and similarly considering these two are squaring we have BC square divided by QR square is equal to AB square divided by PQ square and hence we have proved that area of triangle ABC divided by area of triangle PQR is equal to BC square divided by QR square is equal to AC square divided by PR square is equal to AB square divided by PQ square. Now, let us find the answer to the second part. In the second part we are given the area of triangle ABC divided by area of triangle PQR equal to 9 divided by 16 and we are given BC is equal to 4.5 centimeters we have to find the length of QR. Now, substituting these values here we have 9 divided by 16 is equal to 4.5 centimeters whole square divided by QR square or we have QR square is equal to 4.5 into 4.5 into 16 divided by 9 centimeters square. Now, 9 into 0.5 is 4.5. So, we have 4.5 into 0.5 into 16 centimeters square which further implies 45 into 5 into 16 divided by 10 into 10. Now, in simplifying 5 2's are 10, 2 8's are 16, 5 9's are 45, 5 2's are 10, 2 4's are 8. So, we have 9 into 4 36 centimeters square which implies that QR is equal to 6 centimeters. Hence, the answer is the length of QR is 6 centimeters. So, this completes the session by intake care.