A laser is shot from vertex A of square ABCD of side length 1, towards point P on BC so that BP = 3/4. The laser reflects off the sides of the square, until it hits another vertex, at which point it stops. What is the length of the path the laser takes?
Help?? Won't the laser bounce around until infinity?
No, the laser follows the old billiard table rule, that the angle of incidence is the angle of reflection.
I am going to enlarge your square to a 4 by 4, thus making your first bounce BP = 3
make a reasonable sketch, graph paper might be a good idea.
From P it will bounce and hit CD at Q
look at triangles ABP and PCQ, they are both right-angles and similar.
BP^2 + AB^2 = AP^2
9 + 16 = AP^2
AP = √25 = 5
So each triangle formed by a bounce will have the ratio
3:4:5
We are interested in the sum of the paths formed by the hypotenuses
in the second triangle PC, the short side is 1
so by ratios,
PQ/5 = 1/3
PP = 5/3
find CQ by ratios, then you can find QD
On my diagram, I have the following paths:
From A to P ---- 5 units
from P to Q on CD --- done : 5/3 units
from Q to R on AD , R is close to D,
from R to S on BC
from S to T on AB
from T to U on AD and ahhhh
from U to C , which is a vertex.
At this point you should notice that there is a lot of symmetry
and AP = RS = UC
PQ = TU
QR = TS
so once you have found QR, again by using the ratios 3:4:5
you have found the 3 different path lengths
Add up the 7 lengths, don't forget to divide by 4 , my original step to avoid some initial fractions.
The answer is 5.
Thank you very much it helps a lot!!
So, what is QR? I am a little confused...
For QR I got 3 1/3
Ah, the infinite bouncing laser. It's like a disco party where no one ever leaves the dance floor! But fear not, my friend, in this particular scenario, the laser will eventually hit one of the vertices and come to a stop.
So let's calculate the length of the laser's path. Starting from vertex A, the laser will hit point P, which is 3/4 units away from vertex B. This means it will travel a distance of 3/4 units along BC.
From there, it will reflect off the side of the square and continue towards vertex C. Since the side length of the square is 1 unit, the laser will travel another 1 unit along the side CD.
From vertex C, the laser will then travel towards vertex D, covering yet another 1 unit along the side.
Finally, it will reflect off the side of the square and make its way back to vertex A, completing the square. This adds another 1 unit to the total path length.
So, adding up all the distances, the length of the path the laser takes is 3/4 + 1 + 1 + 1 = 2.75 units.
Hope that sheds some light on the bouncing laser dance party for you!
No, the laser will not bounce around until infinity. In this problem, the laser reflects off the sides of the square until it hits another vertex. Since the square has four vertices, the laser will bounce a maximum of four times before stopping.
To find the length of the path the laser takes, we need to determine the distance traveled during each leg of the path and then add up those distances.
Let's break down the problem step by step:
1. First, let's find the distance from vertex A to vertex P, where the laser hits BC. Given that BP = 3/4 and the side length of the square is 1, we can determine the length of AP using the Pythagorean theorem. Since ABCD is a square, all four sides are equal, so AB = BC = CD = DA = 1. Therefore, AP can be calculated as follows:
AP^2 = AB^2 - BP^2
AP^2 = 1^2 - (3/4)^2
AP^2 = 1 - 9/16
AP^2 = 16/16 - 9/16
AP^2 = 7/16
Taking the square root of both sides: AP = √(7/16) = √7/4
2. Now, let's find the distance from vertex P to the next vertex, Q. The distance PQ is simply the length of the side of the square, which is 1.
3. Next, we need to determine the distance from Q to the next vertex, R. Similar to step 1, we can use the Pythagorean theorem. Since QB = 1 - BP = 1 - 3/4 = 1/4, we can find QR as follows:
QR^2 = QB^2 - BR^2
QR^2 = (1/4)^2 - 1^2
QR^2 = 1/16 - 16/16
QR^2 = -15/16
Since the result is negative, it means that the laser did not hit vertex R.
4. Finally, let's find the distance from R back to A. Since R is not reached, the laser does not travel this leg, and the path ends at vertex R.
Now, let's calculate the total length of the path traveled by the laser:
Path length = AP + PQ + QR
= √7/4 + 1 + QR
= √7/4 + 1 - (did not reach R, so no distance)
Therefore, the length of the path the laser takes is √7/4 + 1.
Note: If the laser had reached vertex R, we would need to find the distance from R back to A and add it to the path length calculation. But since it did not reach R, we stop at vertex R and exclude the last leg from the path length calculation.