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## Geometry (all content)

### Course: Geometry (all content)>Unit 12

Lesson 6: Solving modeling problems with similar and congruent triangles

# Geometry word problem: a perfect pool shot

Sal uses triangle similarity to plan the perfect shot in a pool game. Created by Sal Khan.

## Want to join the conversation?

• At Sal says that the two green angles are congruent because each is complementary to the black angle. How do we know that they are complementary to the black angle? It seems to me that we're just assuming that the line of reflection (the black line between the two blue segmented lines) is perpendicular to right (east) side of the pool, which is not necessarily true.
• The perpendicular is a perpendicular because we have defined it that way. We have just taken the point where the ball hits and purposefully constructed a perpendicular there. In terms of measuring the angles of the path that the ball follows, the only way to "know" how real physical objects behave is to take measurements.

You may have a math theory as to how they behave, but you must test the theory against a real experiment. For instance, in this case, you could take a pool ball, and cover it in wet paint. You could then shoot the ball.

The path that the ball leaves in paint on the table would give clues. You may have a math theory beforehand (a theory based on similar triangles), or you may just perform the experiment, and guess at the math might be afterwards, based on what you see --
You may say to yourself : "the path the ball leaves in paint on the table appears to describe similar triangles : Maybe pool balls bouncing off walls follow a path based on similar triangles" -- and from there , you may draw a perpendicular on the pool table at the point the ball bounced on the wall to see if you seem correct.

Then, you may repeat the experiment numerous times, and test the math against the reality : Test many many many times to feel more confident that the math is really describing what is happening -- and try your best to consider variables which may conflict with your math :
Such as : irregularities in the pool table, the shape of the ball, whether you hit the ball directly in the centre of the ball, etc.) -- but all things considered, if the math seems to generally describe what is occurring physically, over continual / "endless" experiments, there is probably a true relationship between the "reality" and the math theory you are trying to apply to it.

This is one way we can sort of "know", or strongly suspect and feel confident if there is some truth in an idea.
I only know a little tiny bit of high-school entry level physics, but this pool table problem makes me think of "Total internal reflection" in optics, which is to do with similar triangles and a ray being reflected off a surface, and seems quite similar.
See "Snell's Law".
• Hello. Can someone please explain to me @ , how does Sal go from 7/4x to 11/4x?
(1 vote)
• He started with 7/4x + x, got a common denominator 7/4 x + 4/4 x then added 7+4 to get 11/4 x.
• Sometimes saul does the correspondence for sides as (7/4)/1 = ((3/4)-x)/x , while other times he does it as x/1 = ((3/4)-x)/(7/4). When do i use one or another?
(1 vote)
• It doesn't matter. The two are equivalent. Take the first equality and see what happens when you multiply by x/(7/4).
• At , how are the angles complementary to each other? How do you know that mirror line is perpendicular to the rectangle?
• Because the tricky part of the question,
"the angles form as the ball approaches and deflects form a mirror image of each other "

Imagine if we change the ball's position from the original and place it on the north wall center pocket .

We hit the ball and it approached the East wall,
because the question says it form a image,
when the ball leaves the East wall toward South wall, it always creates the exact same angle .

So, the bisector that between these 2 angles will always perpendicular to the East wall .
• I need help on pool problems.
• If you elaborate, I may be able to help!
(1 vote)
• will we get such questions in the GRE examination?
(1 vote)
• They'll not give you these types of problems but the problems that you have to know the logic.
• If two angles of a triangle are equal to each other, does it mean their opposite sites must be equal? And vice versa, please explain it to me.