 Hello and welcome to the session. In this session we are going to discuss the following question and the question says that Draw a dilution of rectangle A, B, C, D on a coordinate plane with A with the coordinates 3, 1, B with the coordinates 3, 2, 5, C with the coordinates 5, 2, 5, D with the coordinates 5, 1. Use original center and scale factor 2 is A, B, C, D similar to A prime, B prime, C prime, D prime. Explain your answer. We know that in dilution if k is scale factor then coordinates x, y changes to kx, ky when center is at origin with this key idea we shall proceed to the solution. In this question we need to draw a dilution of the rectangle A, B, C, D on the coordinate plane. So first we will draw the given rectangle A, B, C, D. In rectangle A, B, C, D we are given vertex A with the coordinates 3, 1, vertex B with the coordinates 3, 2.5, vertex C with the coordinates 5, 2.5 and vertex D with the coordinates 5, 2.5 and vertex D with the coordinates 5, 2.5. So we are given vertex A with the coordinates 3, 1. So we label this point as A, vertex B has coordinates 3, 2.5. So we label this point as B. Similarly vertex C has coordinates 5, 2.5. So we label this point as C and vertex D has coordinates 5, 1. So we label this point as D. Now we join A to B, B to C, C to D and D to A and we get the rectangle A, B, C, D. Now we need to dilute this rectangle and we are given the scale factor is 2. Now from the key idea we know that in dilution if k is the scale factor then the coordinates x, y changes to kx, ky when center is at origin. Now using the above key idea we will find new coordinates by multiplying each coordinate by scale factor 2. So we get A' with the coordinates 3 into 2 that is 6, 1 into 2 that is 2, 6 to B' with the coordinates 3 into 2 that is 6, 2.5 into 2 that is 5. So we get the coordinates 6, 5, C' that is 5 into 2, 10 and 2.5 into 2 is 5. So we get the coordinates of C' as 10, 5 and D' with the coordinates 5 into 2, 10, 1 into 2, 2. So we get the coordinates as 10, 2. Now we plot these points on the coordinate plane. We have point A' with the coordinates 6, 2. So we mark this point as A' we have point B' with the coordinates 6, 5. So we mark this point as B' we have point C' with the coordinates 10, 5. So we mark this point as C' and similarly we have D' with the coordinates 10, 2. So we label this point as D' now we join A' to B' B' to C' C' to D' and D' to A' to get the rectangle A' B' C' D' which is the dilated image of the rectangle A, B, C, D. Now the next part of the question says is A, B, C, D similar to A' B' C' D' that is we have to find whether the two figures are similar or not. First we see corresponding angles since both the figures are rectangles so all angles would be 90 degrees thus we say that angle A is equal to angle A' that is equal to 90 degrees. Similarly angle B is equal to angle D' that is 90 degrees, angle C is equal to angle C' which is equal to 90 degrees and angle D is equal to angle D' that is 90 degrees. So we can say that corresponding angles are same we now see ratio of corresponding sites. First we find the length of sites in original figure and we see that A B is equal to C D is equal to 1.5 here we see that A B is equal to 1.5 and C D is also equal to 1.5. Also A D is equal to B C is equal to 2 here it is clear from the figure that length of A D is equal to 2 units and similarly B C is also equal to 2 units. Now in the dilated figure that is in rectangle A' D' C' D' the length of sites is A' B' is equal to 3 is equal to C' D' we have A' B' is equal to C' D' is equal to 3. Also from the figure we can see that A' D' is equal to 4 is equal to B' C' we have A' D' is equal to B' C' that is equal to 4. Now ratio of the corresponding sites would be given by A' B' upon AB which is equal to 3 by 1.5 will be equal to 30 upon 15 that is 2. Similarly C' D' upon C D is given by 3 upon 1.5 that is 30 by 15 which is equal to 2. The ratio of A' D' to AD is given by 4 by 2 that is equal to 2 and ratio of B' C' to B C is also equal to 4 by 2 that is 2. Here we notice that ratio of all corresponding sites is same so we can say that both the figures are similar but we can say that rectangle ABCD is similar to rectangle A' B' C' D' which is the required answer. This completes our session hope you enjoyed this session.