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cricket20 [7]
2 years ago
6

Use vectors to find the interior angles of the triangle with the given vertices. (Enter your answers as a comma-separated list.

Enter your answers in terms of degrees. Round your answers to two decimal places.) (−3, −5), (2, 6), (6, 3)
Mathematics
1 answer:
SOVA2 [1]2 years ago
4 0

Answer:

23.92°, 78.503°, and 77.577°

Step-by-step explanation:

The coordinates of the vertices of the triangle are;

X(-3, -5), Y(2, 6), Z(6, 3)

The vectors are;

z = \left \langle 2 - (-3), 6 - (-5) \right \rangle = \left \langle 5, 11  \right \rangle

y = \left \langle 6 - (-3), 3 - (-5)  \right \rangle = \left \langle 9, 8 \right \rangle

x = \left \langle 2 - 6, 6 - 3  \right \rangle = \left \langle -4, 3  \right \rangle

cos (\alpha ) = \dfrac{z \cdot y}{\left |  z \right | \times \left | y  \right |}

Therefore, we get;

cos (\alpha ) = \dfrac{5 \times 9  + 11 \times 8 }{\left |  \sqrt{5^2 + 11^2}  \right | \times \left | \sqrt{9^2 + 8^2}   \right |} = \dfrac{133}{\sqrt{146} \times \sqrt{145} } \approx 0.91409

α = arccos(0.91490) ≈ 23.92°

γ = The angle between -y and x

-y = \left \langle -9, -8 \right \rangle

We get;

cos (\gamma) = \dfrac{-y \cdot x}{\left |  -y \right | \times \left | x  \right |}

Therefore;

cos (\gamma) = \dfrac{-9 \times -4  + -8 \times 3 }{\left |  \sqrt{(-9)^2 + (-8)^2}  \right | \times \left | \sqrt{(-4)^2 + 3^2}   \right |} = \dfrac{12}{\sqrt{145} \times \sqrt{25} } \approx 0.199309

γ = arccos(0.199309) ≈ 78.503°

γ ≈ 78.503°

By angle sum property, β = 180° - (α + β)

β ≈ 180° - (23.92° + 78.503°) = 77.577°

β ≈ 77.577°

The interior angles are;

23.92°, 78.503°, and 77.577°

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