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?

Answer 1

To compute the distance between the points, we can apply the distance formula as shown below.

$d = \sqrt{(x_{1} - x_{2})^{2} + (y_{1} - y_{2})^{2} }$

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

$\overline{AABB} = \sqrt{(9-9)^{2} + (12-7)^{2}} = 5$
$\overline{AACC} = \sqrt{(9-1)^{2} + (12-1)^{2}} = \sqrt{185}$
$\overline{BBCC} = \sqrt{(9-1)^{2} + (7-1)^{2}} = 10$

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

$c^{2} = a^{2} + b^{2} - 2abcosC$

or

$C = cos^{-1} (\frac{a^{2} + b^{2} - c^{2}}{2ab})$

Hence, we have

$\angle AA = cos^{-1} (\frac{(\sqrt{185})^{2} + 5^{2} - 10^{2}}{2(5)(\sqrt185)})$
$\angle BB= cos^{-1} (\frac{5^{2} + 10^{2} - (\sqrt{185})^{2}}{2(5)(10)})$
$\angle CC= cos^{-1} (\frac{10^{2} + (\sqrt{185})^{2} - 5^{2}}{2(5)(\sqrt{185})})$

Simplifying this, we have

$\angle AA = 36.03^{0}$
$\angle BB = 126.87^{0}$
$\angle CC = 17.10^{0}$ 

Thus, from this, we can arrange the angles from smallest to largest: ∠CC, ∠AA, and ∠BB.

Answer: ∠CC, ∠AA, and ∠BB