To compute the distance between the points, we can apply the distance formula as shown below.
[tex]d = \sqrt{(x_{1} - x_{2})^{2} + (y_{1} - y_{2})^{2} }[/tex]
In which x₁ and x₂ are the x-coordinates and y₁ and y₂ are the y-coordinates of the two points. Thus, applying this with the segments AABB, AACC, and BBCC, we have
[tex]\overline{AABB} = \sqrt{(9-9)^{2} + (12-7)^{2}} = 5 [/tex]
[tex]\overline{AACC} = \sqrt{(9-1)^{2} + (12-1)^{2}} = \sqrt{185} [/tex]
[tex]\overline{BBCC} = \sqrt{(9-1)^{2} + (7-1)^{2}} = 10 [/tex]
Now that we have the lengths of all the sides of ΔAABBCC, we can find the missing angles using the Law of Cosines.
Generally, we have
[tex] c^{2} = a^{2} + b^{2} - 2abcosC [/tex]
or
[tex] C = cos^{-1} (\frac{a^{2} + b^{2} - c^{2}}{2ab}) [/tex]
Hence, we have
[tex] \angle AA = cos^{-1} (\frac{(\sqrt{185})^{2} + 5^{2} - 10^{2}}{2(5)(\sqrt185)}) [/tex]
[tex] \angle BB= cos^{-1} (\frac{5^{2} + 10^{2} - (\sqrt{185})^{2}}{2(5)(10)}) [/tex]
[tex] \angle CC= cos^{-1} (\frac{10^{2} + (\sqrt{185})^{2} - 5^{2}}{2(5)(\sqrt{185})}) [/tex]
Simplifying this, we have
[tex] \angle AA = 36.03^{0} [/tex]
[tex] \angle BB = 126.87^{0} [/tex]
[tex] \angle CC = 17.10^{0} [/tex]
Thus, from this, we can arrange the angles from smallest to largest: ∠CC, ∠AA, and ∠BB.
Answer: ∠CC, ∠AA, and ∠BB
