Math Exercise - 2026.0624.1953

Setup

2 parellel lines : AB and GC

connecting the line BC

on that line lies a moving point D

connecting the line AD and extend it until it crosses the line GC and end up with an intersection point Q

Question

on the line BC, what should BD be so that the area ABDECDA is the smallest ?

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Solution

add 2 line segments : HD & DF, where

lets name these 4 line segments :

HD as "h1", DF as "h2", CE as "b", BD as "x" which is the length we want to find

lets say the area for ABD is "A1", the area for CED is "A2"

then we can say A = A1 + A2

and thankfully, the 2 triangles ABD and ECD are similar.

with these derivations, we can state the area of the triangles :

and now we can try to come up with a function to calculate the area with "x" as the input

one concept to keep in mind is :

if we were to slide point D from one side to another (i.e. changing x linearly)

the area will shrink then expand (i.e. there exsists a minimum area)

with this in mind, we can find the derivative between the Area and our parameter "x"

when we slide our point, when the area goes from shrinking to expanding, at some point in between, the area must stop shrinking / expanding (i.e. rate of change is 0)

and there we have it. However, to not play with fire, lets select the positive root