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.) (−6,-9), (2,7),(7,2)

Answer 1

23.20°, 71.57°, 85.24°

Explanation:

The angle between two vectors can be found by making use of the definition of the dot product. The computed angle is the angle between the vectors when they are placed tail-to-tail.

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setup

vector definition

The second attachment shows the given points plotted as A, B, C in the order given. In the diagram, the vectors a, b, c are defined as BA, CB, AC, respectively. That is, the vector a is ...

a = A -B = (-6, -9) -(2, 7) = (-6-2, -9-7) = (-8, -16)

The vectors are defined in counterclockwise order around the triangle, though that makes no difference to the calculation.

dot product definition

The definition of the dot product of vectors A and B is ...

A·B = |A|×|B|×cos(θ) . . . . . where θ is the angle between the vectors

Solving for the angle, we find it to be ...

`\theta=\arccos\left(\frac{\vec{A}\cdot\vec{B}}{|\vec{A}|\,|\vec{B}|}\right)`

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computation

Because of the way the vectors in this solution are defined, the angle between any given pair of vectors will be an exterior angle of the triangle. In order to find the measure of the interior angle, we must reverse one of the vectors. That is, the dot product used in our computation will be the opposite of the dot product of the vectors we have found.

The spreadsheet shown in the first attachment does the necessary computations.

• Each vector is the difference of successive points. (x1 -x2, y1 -y2)
• Each dot product shown is the opposite of the dot product of successive vectors. -(x1x2 +y1y2)
• The magnitude is computed in the usual way: the root of the sum of the squares of the components of the vector. √(x²+y²)
• The cosine is the dot product of successive vectors, divided by each of the vector magnitudes.
• The spreadsheet shows the angle in degrees, having converted it from the radian value produced by the ACOS function.

As a check, the spreadsheet shows the sum of the angle values. (The rounded values add up to 180.01°.)

The interior angles of the triangle are 23.20°, 71.57°, 85.24°.

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Additional comment

Another way to find the interior angles from coordinates is to consider the slope of each side of the triangle as being the tangent of the angle it makes with the x-axis. The difference of these angles can be used to find the interior angles of the triangle. Less work is involved because there is no dot-product or magnitude computation.