Key to "Find the measure of each angle of triangle..."

nicekikah

Answered question

2020-12-13

Find the measure of each angle of triangle ABC, given AC = 6,

C B = 2 \sqrt 3

, and

A B = 4 \sqrt 3 .

Ayesha Gomez

Skilled2020-12-14Added 104 answers

Use the Law of Cosines to find two angle measures and the Triangle Sum Theorem to find the third angle.
Find $∠ A :$

$C B ² = A C ² + A B ² - 2 (A C) (A B) \cos ∠ A$

$2 (A C) (A B) \cos ∠ A = A C ² + A B ² - C B ²$

$\cos ∠ A = \frac{A C ² + A B ² - C B ²}{2 (A C) (A B)}$

$\cos ∠ A = (6 ² + (4 \sqrt{3}) ² - \frac{2 \sqrt{3}) ²}{2 (6) (4 \sqrt{3})}$

$\cos ∠ A = \frac{72}{48} \sqrt{3}$

$\cos ∠ A = \frac{3}{2} \sqrt{3}$

$\cos ∠ A = \sqrt{\frac{3}{2}}$

$∠ A = \cos^{-} 1 \times (\sqrt{\frac{3}{2}})$

$∠ A = 30 °$

Find $∠ B$

$A C ² = C B ² + A B ² - 2 (C B) (A B) \cos ∠ B :$

$2 (C B) (A B) \cos ∠ B = C B ² + A B ² - A C ²$

$\cos ∠ B = \frac{C B ² + A B ² - A B ²}{2 (C B) (A B)}$

$\cos ∠ B = \frac{(2 \sqrt{3}) ² + (4 \sqrt{3}) ² - 6 ²}{2 (2 \sqrt{3}) (4 \sqrt{3})}$

$\cos ∠ B = \frac{24}{48}$

$\cos ∠ B = \frac{1}{2}$

$∠ B = \cos^{-} 1 \times (\frac{1}{2})$

$∠ B = 60 °$

Find $∠ C$ using Triangle Sum Theorem:

$∠ A + ∠ B + ∠ C = 180 °$

$30 ° + 60 ° + ∠ C = 180 °$

$∠ C = 90 °$

Jeffrey Jordon

Expert2021-08-10Added 2605 answers

Answer is on video

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