Problem 30
The vertices A, B, and C, of an isosceles triangle ABC, where AB = AC, lie on three parallel lines, l, m, and n respectively. If the distance between l and m is 28 and that between m and n is 8, where m lies between l and n, find the minimum area of ΔABC.
Here’s a simulation that illustrates the problem.
Try dragging A or B. Then, C will move such that AB = AC. When is the area of ΔABC minimum? How can we prove the same?
Solution
There are quite a few different ways in which you can solve this problem. I’ll help you get started with a few of them. In each case, we’ll find the area of the triangle in terms of a single variable, and try to minimize the expression obtained.
Method 1: Trigonometry
[coming soon]
Then, the area of the triangle equals:
(1/2).2x.y.sin(α+β) or xysin(α+β)
Method 2: Calculus
[coming soon]
Were you able to solve it using any of the methods given above? Or, did you solve using any other method? Or, do you need some more help? You can reach out at the Telegram group.
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