Finding a quadrilateral from its symmetries

two point two of the points that define a certain quadrilateral are zero comma nine and three comma four the quadrilateral is left unchanged by a reflection over the line y is equal to three minus x draw and classify the quadrilateral now I encourage you to pause this video and try to draw and classify it on your own before I'm about to explain it so let's at least plot the information they give us so the point zero zero comma nine that's one of the vertices of the quadrilateral so zero comma nine that's that point right over there and another vert vert is another one of the vertices is three comma four three comma four that's that right over there and then they tell us that the quadrille is left is left unchanged by reflection over the line y is equal to three minus x so when X is zero Y is three that's our y-intercept and it has a slope of negative one you could view this as three minus one X so it has a slope of negative one so the line looks like this so every time we increase our X we buy one we decrease our y by one so the line looks something like this y is equal to three minus x try to draw it relatively very pretty carefully pretty carefully so that's what it looks like Y is equal to three minus x so that's my best attempt at drawing it y is equal to three minus X so the quadrilateral is left unchanged by a reflection over this so that means if I were to reflect each of these vertices I would essentially end up with one of the other vertices on it and if those get reflected you're going to end up with one of these so the thing is not going to be different so let's think about where these other two vertices of this quadrilateral need to be so this point let's just reflect it over this line over Y is equal to three minus X so if we were to try to drop a perpendicular to this line notice we have gone diagonally across one two three of these squares we need to go diagonally across three of them on the left hand side so one two three gets us right over there that's the reflection this point this is the reflection of this point across that line now let's do the same thing for this blue point to go to drop a perpendicular to this to this line we have to go diagonally across two of these squares so let's go diagonally across two more of these squares just like that to get to that point right over there and now we've defined our quadrilateral our quadrilateral looks like this our quadrilateral looks like this both of these lines are perpendicular to that original line so they're going to have the same slope so that line is parallel to that line over there and then we have this line we have this line and then we have this line so what type of quadrilateral is this well I have one pair of parallel one pair of parallel sides so this is a this is a trapezoid