 Hello and welcome to the session. In this session, we shall discuss if one figure is obtained from the other figure using translation, reflection, rotation and dilation, then the obtained figure is similar to the given figure. Now we know that two figures are similar if the corresponding sides are proportional and they have same corresponding angles that is here AB upon DE is equal to BC upon EF is equal to AC upon DF. So, in similar triangles the corresponding sides are proportional and the corresponding angles are equal that is angle A is equal to angle D, angle B is equal to angle E and angle C is equal to angle F. And in congruent figures corresponding sides are equal and the corresponding angles are also equal. So, here these two triangles are congruent it means they have same shape and size whereas these three triangles are similar. So, they have same shape but their size can vary as here the corresponding sides are proportional. You must note one thing that shape of two figures can be known with the help of corresponding angles and size can be known by the length of corresponding sides of two figures. So, if corresponding angles of two figures are equal then they have same shape. Now let us discuss an example. Now in this example we have to check that whether the figure ABCD is similar to the figure PQRS. For this we will measure the corresponding angles. Now here angle A and P, angle D and S then angle R and angle C then angle Q and angle B are corresponding angles. So, we will measure these angles and then we will find the ratio of the corresponding sides. Now on this graph paper one square block represents one unit. So, we can measure the sides of two figures. Now in figure ABCD length of each side is four units and in figure PQRS is of two units. Also in both figures all sides are straight lines horizontal or vertical forming 90 degree angle with each other. So, all angles are of 90 degrees. Now we know that if all angles and all sides of a quadrilateral are equal then that quadrilateral is a square. So, here the quadrilateral ABCD and the quadrilateral PQRS are squares. Now here the corresponding angles are equal. Now let us find the ratio of corresponding sides. So, here ratio of each corresponding side is equal to two it means corresponding sides of the two figures are proportional. So, in these two figures corresponding angles are equal corresponding sides are proportional the two figures are similar. Now in our earlier sessions we have learnt about various forms of transformations and those transformations were dilation, translation, reflection and rotation. Now dilation enlarges or contracts a figure from a fixed centre by a common scale factor. Now here the triangle ABC is enlarged to triangle DEF with O as the centre. Now here as the figure is enlarged so there is a change in size but shape remains same so a dilated figure is similar to its preimage that is here the figure ABC is similar to the figure DEF. Now translation moves the object so that every point on the object moves in the same direction as well as same distance. Now here the triangle DEF is translated to the triangle PER and here the point D has moved to point P, point E has moved to point Q and point F has moved to point R. So in translation the figure slides it means that in translation the shape and size of both the figures remain same. Now as shape and size of both the triangles remain same it means corresponding angles and corresponding sides of both the triangles are equal. So here triangle DEF is congruent to triangle PQR. Now in reflection an object is flipped across a line of reflection so shape and size remain same and in rotation an object is only rotated by some angle so here also the shape and size of figure remains same. So we obtain similar figures if we combine dilation with translation or reflection or rotation. Thus a figure in a plane is similar to another figure if second can be obtained from the first by sequence of translations, rotations, reflections and dilations because when the size will vary that shape remains same. We can also dilate, translate, rotate and reflect together in the sequence as asked. Now let us discuss an example. In this example we have to describe the sequence of transformations that result in transformation of figure A to figure A dash. Now here figure A is the triangle with vertices P, Q and R and coordinates of P are minus 2 minus 2, coordinates of Q are 0, 0 and coordinates of R are 2 minus 2. Now here the figure A dash is again a triangle with vertices P double dash, Q double dash, R double dash and coordinates of P double dash are 3 3, coordinates of Q double dash are 4 2 and coordinates of R double dash are 5 3. Now here we can observe that figure A dash is small in size and its position is changed also it seems to be a reflection in x axis. So first figure that is the figure A is dilated then it is translated and then it is reflected in x axis to obtain the figure A dash. So figure will be contracted if scale factor is less than 1. Now first of all let us find the scale factor. Now we know that scale factor is equal to drawing dimension upon actual dimension. Now here the scale factor will be equal to dimension of figure A dash upon dimension of figure A. Now let us take the length of P double dash R double dash which is 2 units upon length of P R which is 4 units. So scale factor is equal to 1 upon 2 which is less than 1 so figure A will be contracted. So triangle P Q R is contracted to form a triangle Q X Y and coordinates of X are minus 1 minus 1, coordinates of Y are 1 minus 1 and coordinates of Q are 0 0. Now see the figure A dash here Q double dash is 2 units above x axis and P double dash and R double dash are 3 units above x axis. So the reflection of Q double dash will be 2 units below the x axis. So this is the point which is the reflection of the point Q double dash and let us name it as Q dash. Similarly this point is reflection of P double dash and this point is reflection of R double dash. Now on joining these points we get a triangle P dash Q dash R dash which is the reflection of the triangle P double dash Q double dash R double dash along the x axis and these are the coordinates of P dash Q dash R dash. Now if we translate the triangle Q X Y 4 units to the right and 2 units down then we get the triangle P dash Q dash R dash. So for translation we will add 4 to the x coordinates and we will subtract 2 from the y coordinates of the vertices of triangle Q X Y and then after translation we obtain a new triangle P dash Q dash R dash and then if we reflect this triangle P dash Q dash R dash along the x axis then we get the figure A dash. So first of all the triangle P Q R is dilated to get a new triangle Q X Y and then this triangle is translated to get a new triangle P dash Q dash R dash and finally after reflection along the x axis we get the triangle P double dash Q double dash R double dash which is the figure A dash. So in this session we have learned about similarity of figures using transformation and this completes our session. Hope you all have enjoyed the session.