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# Geometry word problem: a perfect pool shot

CCSS Math: HSG.SRT.B.5

## Video transcript

A pool table is 1 meter by 2 meters. So this is 1 meter-- let me label that-- so this distance right over here is 1 meter. This distance right over here is 2 meters. That's 2 meters right over there. There are 6 pockets total, 1, 2, 3, 4, 5, 6, 4 in the corners and 2 at the midpoints of each of the 2-meter sides, or the midpoints of each 2-meter side. A cue ball is placed 0.25, or a quarter meter, from the north wall and a quarter meter away from the west wall, so this right over here, this distance right over here. So it's a quarter meter away from the north wall. So this is 1/4 meter. And this distance right over here is also 1/4 of a meter. So that's that distance, that's that distance. From the west wall. The angles formed as the ball approaches and deflects form a mirror image of each other. So this is as we approach, and then we deflect. And they're mirror images. So if we were to draw-- if we could imagine a mirror right over there-- we see that they are mirror images of each other. At what distance x-- so then they labeled x right over here-- from the southeast corner should the cue ball hit the east wall-- so this is the distance from the southeast corner-- so the cue ball sinks into the pocket at the midpoint of the south wall? And I encourage you to pause the video. And I'll give you one hint. It might involve similar triangles. So let's try to work it through. So a big clue is that the approach and the deflection, that they're going to be mirror images of each other. So if they're mirror images, this angle is going to be congruent to that angle. If those two angles are congruent, then this angle, which is complementary to that black angle, must be equal to this angle. Each of these are going to be 90 degrees minus that black angle right over there. So you have this angle is congruent to that angle. And now we can construct two right triangles. So you can imagine one up here, this is the larger one. So imagine this is our approach right triangle right over here. So the top of it is parallel to the side of the pool table. And then this is our deflection right triangle right over here. And the whole reason why I show that these two green angles are going to be congruent is to show that these two triangles are similar to each other. How do we know they're similar? Well, if you have two angles, they both have a 90 degree angle, and they both have this green angle, so that means that the third angle must also be the same. If you know two angles, you know what the third angle has to be. If two corresponding angles of two different triangles are congruent, then the triangles are going to be similar. So this top triangle is similar to this bottom triangle. And what that helps us is that means that the ratio of the lengths of corresponding parts of those triangles are going to be the same. So for example, we've already said that this distance-- let's figure out what we know about these triangles-- so that distance is x. Now, what is this distance right over here? What is this distance right over here going to be? Well, let's think about it a little bit. We know that this distance is 1/4 of a meter. We know that this entire distance is 1 meter. So this distance right-- let me do this in a color you can see-- this distance right over here is going to be 3/4 of a meter. And so if this distance is 3/4 of a meter, then this part right over here is going to be 3/4 minus x meters. So let me write that down. 3/4 minus x is this magenta length. And what else do we know? Well, we definitely know the length of this segment right over here. We know that the pockets are 1 meter apart, so that is 1 meter. And we also know the length of this segment right over here. We know that this is 1 meter and that this is another 3/4 of a meter. This whole distance right over here is 1 and 3/4 meters. Or we could write that as 7/4 meters. So let me write it like this. This is 7/4. I like to write everything as an improper fraction because I have a feeling that I'm going to have to do some ratios in a second. So corresponding parts of these triangles-- these two triangles are similar, so corresponding parts are going to have the same ratio. So for example, this green segment right over here, this is the longer side that's not the hypotenuse of this top right triangle. That's going to correspond to the longer side that's not the hypotenuse of this triangle. The sides that are opposite this green angle, they correspond to each other. So we could say that the ratio of 7/4 to 1, the ratio of 7/4 meters to 1 meter, is going to be equal to the ratio of the sides that are opposite the magenta angles. So it's going to be equal to 3/4 minus x to x. I'm just showing that the ratio of corresponding parts are the same. So let's now just solve for x. So let's see, if we multiply both sides of this by x, on the left hand side, we end up with 7/4 x. And on the right hand side, we're left with 3/4 minus x. Well, now we can add an x to both sides. And we're going to get 7/4 x plus another 4/4 x is going to give us 11/4 x is equal to 3/4. And now to solve for x, we can just multiply both sides times the reciprocal of its coefficient, so times 4/11. This is going to give us that x is equal to 3/11 of a meter. So if we hit 3/11 of a meter above the southeast corner of this wall, then we should hit this pocket right over here.