Explanation
Length of CD
From the picture, we know two sides and an angle of the triangle CDE. We define the sides and angle:
• a = EC = 440.68,
,
• b = ED = 470.43,
,
• c = CD = ?,
,
• γ = 60° 06' 09''.
From trigonometry, we know that the Law of Cosines states that:
[tex]\begin{gathered} c^2=a^2+b^2-2ab\cdot\cos\gamma, \\ c=\sqrt{a^2+b^2-2ab\cdot\cos\gamma}. \end{gathered}[/tex]
Where the angle γ and the sides a, b and c are defined by:
Replacing the values from above in the equation for side c, we get:
[tex]c=\sqrt{(440.68)^2+(470.43)^2-2\cdot440.68\cdot470.43\cdot\cos(60\degree06^{\prime}09^{\prime}^{\prime})}\cong457.10.[/tex]
Length of AB
To compute the length of AB, first, we must compute the length of sides AE and EB.
Side EB
From the picture, we see a triangle ECA. Using the data of the picture, we have:
• EC = 440.68,
,
• ∠E = 60° 06' 09'',
,
• EA = ?,
,
• ∠A = ?.
,
• ∠C = 97° 17' 42''.
Angles ∠A, ∠E and ∠C are the inner angles of triangle ECA, so they must sum up 180°, so we have:
[tex]\begin{gathered} ∠A+∠E+∠C=180\degree, \\ ∠A=180\degree-∠E-∠C, \\ ∠A=180\degree-60\degree06^{\prime}09^{\prime\prime}-97\degree17^{\prime}42^{\prime\prime}=22°36^{\prime}9^{\prime\prime}. \end{gathered}[/tex]
Now, we define the following sides and angles:
• c' = EC = 440.68,
,
• γ' = ∠A = 22° 36' 9''
,
• a' = EA = ?,
,
• α = ∠C = 97° 17' 42''.
Now, from trigonometry, we know that the Law of Sine states that:
Using the equation that relates a' and c', we have:
[tex]\begin{gathered} \frac{a^{\prime}}{\sin\alpha^{\prime}}=\frac{c^{\prime}}{\sin\gamma^{\prime}}, \\ a^{\prime}=c^{\prime}*\frac{\sin\alpha^{\prime}}{\sin\gamma^{\prime}}. \end{gathered}[/tex]
Replacing the values from above, we get:
[tex]EA=a^{\prime}=440.68*\frac{\sin(97°17^{\prime}42^{\prime\prime}^)}{\sin(22°36^{\prime}9^{\prime\prime})}[/tex]
Side AE
From the picture, we see a triangle EDB. Using the data of the picture, we have:
• b' = ED = 470.43,
,
• ∠E = 60° 06' 09'',
,
• a' = EB = ?,
,
• α' = ∠D = 180° - 87° 20' 24'' = 92° 39' 36'',
,
• β' = ∠B = 180° - ∠D - ∠E = 180° - 92° 39' 36'' - 60° 06' 09'' = 27° 14' 15''.
Applying the law of sines, we have that:
[tex]\begin{gathered} \frac{a^{\prime}}{\sin(\alpha^{\prime})}=\frac{b^{\prime}}{\sin(\beta^{\prime})}, \\ EB=a^{\prime}=b^{\prime}*\frac{\sin(\alpha^{\prime})}{\sin(\beta^{\prime})}. \end{gathered}[/tex]
Replacing the values from above, we get:
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Answer
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