Mathematics

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

  • 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
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