Erica plotted the three towns closest to her house on a graph with town AA at (9, 12), town BB at (9, 7) and town CC at (1, 1). She drew the triangle joining th
Mathematics
emilybrdn
Question
Erica plotted the three towns closest to her house on a graph with town AA at (9, 12), town BB at (9, 7) and town CC at (1, 1). She drew the triangle joining the 3 points. Which lists the angles formed in size, smallest to largest?
1 Answer

1. User Answers sharmaenderp
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 xcoordinates and y₁ and y₂ are the ycoordinates of the two points. Thus, applying this with the segments AABB, AACC, and BBCC, we have
[tex]\overline{AABB} = \sqrt{(99)^{2} + (127)^{2}} = 5 [/tex]
[tex]\overline{AACC} = \sqrt{(91)^{2} + (121)^{2}} = \sqrt{185} [/tex]
[tex]\overline{BBCC} = \sqrt{(91)^{2} + (71)^{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