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6.4504 < x < 10

Step-by-step explanation:

The solution to this related-lengths problem can involve as much depth as you like.

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assertions

Here, we will assert that the minimum length of segment CD will occur when AB = AC = AD. That is, the two triangles are both isosceles.

The maximum possible length of CD will depend on the additional constraints you add to the problem. If we assume that point C cannot be to the left of point A, then CD will be a maximum when AC = 0.

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minimum CD

When ΔBAC and ΔDAC are both isosceles, with AB = AC = AD, we can define altitudes AE and AG to midpoints E and G of BC and DC, respectively. The relations between these segment lengths is ...

CE/CA = sin(68°/2) = sin(34°)

CG/CA = sin(32°/2) = sin(16°)

Solving this pair of equations for CG, we get ...

CG = CE·sin(16°)/sin(34°)

Then the lengths of interest are ...

CD = 2·CG = 2·CE·sin(16°)/sin(34°) = 77·sin(16°)/sin(34°)

11x -33 = 77·sin(16°)/sin(34°)

11x = 77·sin(16°)/sin(34°) +33

x = (77·sin(16°)/sin(34°) +33)/11

x = 7·sin(16°)/sin(34°) +3 ≈ 6.4504 . . . . minimum x

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maximum CD

When points A and C are coincident, lengths AB and AD are 77. This means ...

CD = 77

11x -33 = 77

x -3 = 7 . . . . . . . divide by 11

x = 10 . . . . . . maximum x

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The possible range of x is ...

6.4504 < x < 10

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The assertion that CD is a minimum when AB=AC=AD can be shown to be true using the method of Lagrange multipliers. The Lagrangian for the geometry can be written using the law of cosines:

ℒ = AB² +AC² -2·AB·AC·cos(32°) +λ(AB² +AC² -2·AB·AC·cos(68°) -77²)

The partial derivatives are zero:

∂ℒ/∂AB = 0 = 2(AB -AC·cos(32°) +λ(AB -AC·cos(68°)))

∂ℒ/∂AC = 0 = 2(AC -AB·cos(32°) +λ(AC -AB·cos(68°)))

∂ℒ/∂λ = 0 = AB² +AC² -2·AB·AC·cos(68°) -77²

These equations have solutions ...

AC = ±AB

AC = ∓77/√(2(1±cos(68°))) ≈ {-46.4394, 68.8492}

In the case for AC = -46.4394, point C lies to the left of point A, and ΔDAC is obtuse. CD is 77·cos(16°)/cos(34°) ≈ 89.2808, and the maximum value of x is about 11.1164.

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It doesn't seem like the intent of this problem is to require this level of analysis and use of calculus and trigonometry. We suspect simpler assumptions are expected regarding the possible angles and/or segment lengths—assumptions that would give rational results.
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