[Music] welcome to video one for this unit on dilations similarity and introducing slope in this video we're going to focus on dilations here is a point p and triangles a b and c what do you notice about these triangles you may have noticed that the three triangles seem to have the same three angles but are different sizes let's add some lines to make the relationship between the three triangles and point p clearer earlier we learned about three rigid transformations translations rotations and reflections this is an example of a different type of transformation that isn't rigid it is called a dilation here is what a dilation of triangle b to triangle a looks like it's like and here is what a dilation of triangle b to triangle c looks like a dilation is a transformation in which each point on a figure moves along a line and changes its distance from a fixed point the fixed point is the center of the dilation all of the original distances are multiplied by the same scale factor here is a circular grid the radius of the smallest circle is one unit and the radius of each successive circle is one unit more than the previous one a circular grid like this one can be helpful for performing dilations here is triangle a b c and triangle a prime b prime c prime triangle prime c prime is a dilation of triangle a b c to perform a dilation we need a center of dilation a scale factor and a point on the original figure to dilate in the picture p is the center of dilation the center of a dilation is a fixed point on a plane it is the starting point from which we measure distances in a dilation in this example the scale factor of the dilation is 2. that means that each corresponding point stays in the same ray from p but its distance from p doubles that is since the circles on the grid are the same distance apart segment p a prime has length 6 units which is double the length of segment pa let's look at a dilation done without a grid here point a is the center of the dilation if we were using a circular grid point a would be in the middle and points b c and d would be on the same line how can we find which point is the dilation of b with scale factor two since the scale factor is larger than one the point must be further away from a than b is which makes c the point we are looking for if we measure the distance between a and c we would find that it is exactly twice the distance between a and b a dilation with scale factor less than one brings her to the center of dilation the point d is the dilation of b with center a and scale factor one-third that is the distance from a to d is one-third the distance from a to b square grids can be useful for showing dilations the grid is helpful especially when the center of dilation and the points being dilated lie at grid points rather than using a ruler to measure the distance between the points we can count grid units for example suppose we want to dilate point q to point q prime with center of dilation p and scale factor three halves earlier we used a ray starting at the center of dilation to figure out the location of the image after the dilation however we can use a square grid to show a dilation as well pointl point q is four grid squares to the left and two grid squares down from p where will point q prime be on the grid since three halves is larger than one point q prime will be further from point p than point q is using the structure of the grid we can scale the horizontal and vertical distances from point p to point q by the scale factor of three halves to find the location of point q prime since four times three halves is six and two times three halves is three point q prime is six grid squares to the left and three grid squares down from point p sometimes the square grid comes with coordinates the coordinate grid gives us a convenient way to name points and sometimes the coordinates of the image can be found with just arithmetic for example to make a dilation with center 0 comma 0 and scale factor 2 of the triangle shown let's start by listing the coordinates of its vertices negative 1 comma negative two three comma one and two comma negative one to find the coordinates of the new triangle we can double these coordinates to get negative two comma negative four six comma two and four comma negative two now that we know the coordinates of the vertices of the dilated triangle we can draw in the corresponding segments the one of three for this unit on dilations similarity and introducing slope you