Determine the interior angles of triangle ABC for a(5,1) b(4,-7) and c(-1,-8)

Please type your subject in the School Subject box. Any other words, including obscure abbreviations, words, or phrases, are likely to delay responses from a teacher who knows that subject well.

I will use AB for vector AB ... etc

AB=(-1,8)
AC=(-6,7)
AB•AC = |AB||AC|cosA
6+56 = √65√85cosA
cosA = 33.48°

Find a second angle the same way..
Use the 180° angle sum to find the third.

To determine the interior angles of triangle ABC, we can use the concept of vector dot product and trigonometry. Here's how you can find the interior angles:

1. Find the vectors of two sides of the triangle: AB and BC.
- Vector AB = Vector B - Vector A
AB = (4, -7) - (5, 1) = (-1, -8)
- Vector BC = Vector C - Vector B
BC = (-1, -8) - (4, -7) = (-5, -1)

2. Use the dot product formula to find the angle between two vectors.
- Dot product formula: A · B = |A| × |B| × cos(θ)
- The dot product of two vectors is equal to the product of their magnitudes and the cos of the angle between them.

3. Find the magnitudes (lengths) of the vectors AB and BC.
- Magnitude AB = sqrt((-1)^2 + (-8)^2) = sqrt(1 + 64) = sqrt(65)
- Magnitude BC = sqrt((-5)^2 + (-1)^2) = sqrt(25 + 1) = sqrt(26)

4. Calculate the dot product of the vectors AB and BC.
- AB · BC = (-1) × (-5) + (-8) × (-1) = 5 + 8 = 13

5. Use the dot product and trigonometry to find the angle between AB and BC.
- AB · BC = |AB| × |BC| × cos(θ)
- cos(θ) = (AB · BC) / (|AB| × |BC|)
- cos(θ) = 13 / (sqrt(65) × sqrt(26))
- θ = arccos(13 / (sqrt(65) × sqrt(26)))

6. Convert the angle from radians to degrees, if necessary.

By following these steps, you can find the interior angle between any two sides of triangle ABC. Repeat the process for the other angles.