Problem 80 Hard Difficulty

A right triangle $ ABC $ is given with $ \angle A = \theta $ and $ \mid AC \mid = b. $ $ CD $ is drawn perpendicular to $ AB, DE $ is drawn perpendicular to $ BC, EF \bot AB, $ and this process is continued indefinitely, as shown in the figure. Find the total length of all the perpendiculars
$ \mid CD \mid + \mid DE \mid + \mid EF \mid + \mid FG \mid + \cdot \cdot \cdot $
in terms of $ b $ and $ \theta. $

Answer

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Video Transcript

so right Triangle ABC is given. We're given angle, eh? We have some angles Data here. We're given the length of Decide A C is equal to B and then we draw CD this perpendicular line. Teo. Maybe that's why we have this right angle here and then we go in and draw d e. So that is perpendicular to BC. That's what we have the right angle and we would just continue this process. For example, after this point, we would go perpendicular early to a B and then we would meet at some point f and we would just continue this process forever. This will happen infinitely many often. The process is continued indefinitely In the figure shown below, I find this whole length of the perpendicular Sze CD D Basically all these lines here not not the original a C that's not included on ly The red lines are erratic, so let's just find the pattern here Will see that eventually that this is a geometric so seedy Well, this is a right triangle over here. This a DC This is a race, rangel. So you could use your right triangle trigonometry here to see that Well, we have signed data equals the Red Line, which is CD. Let's just overby opposite over hypotenuse so I can rewrite this by solving for CD. I get be signed data. So this first this largest red line here Yeah, CD is equal to be signed, Ada, and that's what we want. We want to first express all of these red lines in terms of being data and then we'LL come back and add the sum. So first, we're just trying to see what kind of sum it is. So let's go and find a few these red lines. The next one would be the so this time you are looking at the triangle the E C. So triangle T e c. This is another right triangle. And if this angle was data soul goes, go back to Triangle ABC. This was angle data. That's a ninety degree angle. That means this angle here is ninety minus data. But that must mean that this angle down here is stada. So we have another right triangle in here. We have our data so we can go ahead and say that the sign of data is equal to CD the length of CD over the length D Just using definition of sign The sign is thie opposite over high ponders. So that's d over the iPod news. And here I I made one mistake there. I'm sorry about that. So let me take a step back here. I have the order incorrect. That should be the over CD because again, opposite over high pounders. So solve this for the you and we get that it equals a CD signed data. But then using the previous part, this's be sine squared data, and we'll eventually see that this is the pattern. Each time we find us a progressive a progression so the next one would be e f. This one will be signed data times E. So this one will be be signed Cube data. So then this infinite sum here that we're being asked to compute that's just equal to be signed data be signs where there be signed cubes, ada and so on. And this is a geometric series because we're multiplying by the same term each time which is signed data. And we know that this sign satisfies the inequality that is between less than one in one. So we know, sometimes I can be negative. One side can be one. But based on this picture, here we are, assuming that this is not ninety degrees. So here this is angle between zero and pirates do. And we know if this is our assumption that the sign will just be between zero and one. So this will converge. And then we just final answer is to use the formula for the Convergence Geometric series. It's the first term of the series. And then you just defined by one minus our sonar problem. The first term is be signed data. We can see that that was the length of the first term. The first side, the longest one over here. And then we just defined by our one minus r. And we've said our was signed data. So one bite. This signed data. And that's our final answer. That's the sum of the length of the perpendicular Sze