Problem 15
Let ABCD be a square, whose side length is x. Let P and Q be points on AB and AD respectively, such that ∠CPQ = 90º. For what position of P will AQ be of maximum length? What’s this maximum length in terms of x?
Here’s a simulation that demonstrates the problem.
You can drag the point P. The point Q will move such that ∠CPQ = 90º. When is AQ of maximum length? How can we prove that?
Solution
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.
Range of a Quadratic Expression
Consider the quadratic expression ax2 + bx + c, where x is a real number. Then,
if a > 0, the minimum value of the expression is (4ac – b2)/4a
if a < 0, the maximum value of the expression is (4ac – b2)/4a
To proceed, try finding similar triangles in the given figure. Then, using the proportionality of sides of these triangles, express AQ in terms of the known lengths. You should get a quadratic expression involving a variable. Try finding its maximum value.
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.
Problem Source: CutTheKnot
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