Problem 10
Two circles, S and S’, intersect each other at two points, P and Q. A point A is chosen on S and AP is joined and extended, meeting S’ at B.
For what position of A will AB be maximum?
Here’s a simulation that demonstrates the problem.
Try dragging the point A. At what position of A will AB be of the maximum length? How can you prove that?
Solution
I won’t provide the full solution here, but only give you some tools and hints that’ll help you solve the problem yourself. Here’s what you’ll need to solve the problem.
Similar Triangles
If two triangles have the same ‘shape’, then they are said to be similar. This means that the corresponding angles of these triangles are equal. Also, their corresponding sides are proportional.
For example, if ∆ABC and ∆DEF are similar, then
A = D, B = E, C = F
AB/DE = BC/EF = CA/FD
Here’s a simulation that shows two similar triangles. You can drag the vertices and observe the angles and side lengths.
To prove that two triangles are similar, you can prove that their corresponding angles as equal. Then, you can use the fact that their corresponding sides are proportional. Or, vice versa.
Maximizing Terms in a Ratio
If two quantities are in a given ratio, then if one of them is increased, the other will also increase. Similarly, if one of them decreases, the other will also decrease (to maintain that ratio).
This means that to maximize (or minimize) one of them, you have to maximize (or minimize) the other as well.
In mathematical terms, if a : b is fixed, then b will be maximum when a is maximum. Similarly, b will be minimum when a is minimum.
To proceed, you need to use the ratio of the sides of similar triangles (which are invisible as of now), and see if maximizing something else leads to maximization of AB.
That’s all for now. I suggest you proceed on your own from this point. Good luck!
Were you able to solve it using the above hints? Or, did you solve using any other method? Or, do you need some more help? You can reach out at the Telegram group.
Source: CutTheKnot
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