Question

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.)

(−2, 4), (−3, 8), (6, 8)

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Answer 1

77.47°

75.96°

26.57°

Explanation:

Given vertices of the triangle:

• A = (−2, 4)
• B = (−3, 8)
• C = (6, 8)

Find the vectors from A to B, B to C and A to C:

`\begin{aligned}AB = B - A &=(x_B-x_A,y_B-y_A) \\&=(-3-(-2), 8-4)\\& = (-1, 4) \end{aligned}`

`\begin{aligned}BC=C-B &=(x_C-x_B,y_C-y_B)\\ &=(6-(-3),8-8)\\&=(9,0)\end{aligned}`

`\begin{aligned}AC = C - A &=(x_C-x_A,y_C-x_A)\\&= (6-(-2), 8-4) \\&= (8, 4)\end{aligned}`

Use Pythagoras Theorem to calculate the magnitudes of the vectors:

`|AB| = √((-1)^2+4^2)=√(17)`

`|BC|=√(9^2+0^2)=9`

`|AC| = √(8^2+4^2)=4√(5)`

`\boxed{\begin{minipage}{6 cm}\underline{Dot Product of two vectors}\\\\$a \cdot b=|a||b| \cos \theta$\\\\where:\\ \phantom{ww}$\bullet$ $|a|$ is the magnitude of vector a. \\ \phantom{ww}$\bullet$ $|b|$ is the magnitude of vector b. \\ \phantom{ww}$\bullet$ $\theta$ is the angle between $a$ and $b$. \\ \end{minipage}}`

Rearrange the dot product formula to make θ the subject:

`\implies \theta=\cos^(-1)\left((a \cdot b)/(|a||b|)\right)`

Use the rearranged dot product formula to find the angles between two pairs of vectors.

`\boxed{\begin{minipage}{4 cm}\underline{Dot Product}\\\\$\textbf{u} \cdot \textbf{v}=u_1v_1+u_2v_2$\\\\where:\\ \phantom{ww}$\bullet$ $\textbf{u}=\left\langle u_1,u_2 \right\rangle$ \\\phantom{ww}$\bullet$ $\textbf{v}= \left\langle v_1,v_2 \right\rangle$ \\ \end{minipage}}`

Angle A

`\implies A=\cos^(-1)\left((AB \cdot AC)/(|AB||AC|)\right)`

`\implies A=\cos^(-1)\left((-1 \cdot 8+4 \cdot4)/(√(17) \cdot 4 √(5))\right)`

`\implies A=\cos^(-1)\left((8)/(4 √(85))\right)`

`\implies A=77.47^(\circ)\; \sf (2 \; d.p.)`

Angle C

`\implies C=\cos^(-1)\left((BC \cdot AC)/(|BC||AC|)\right)`

`\implies C=\cos^(-1)\left((9 \cdot 8+0 \cdot4)/(9 \cdot 4 √(5))\right)`

`\implies C=\cos^(-1)\left((72)/(36 √(5))\right)`

`\implies C=26.57^(\circ)\; \sf (2 \; d.p.)`

Interior angles of a triangle sum to 180°.

`\implies B=180^(\circ)-A-C`

`\implies B=180^(\circ)-77.47^(\circ)-26.57^(\circ)`

`\implies B=75.96^(\circ)`

Therefore, the interior angles of the triangle with the given vertices are:

• 77.47°
• 75.96°
• 26.57°