Question

△ABC has vertices A(−1, 2), B(2, 8), and C(4, 1). Find the measure of each angle of the triangle. Round decimal answers to the nearest tenth.

Answer 1

To find each angle in triangle ABC, one must calculate the lengths of the sides using the distance formula and then use the Law of Cosines. Applying this method, you can find the cosines of the angles and then their measures by taking the inverse cosine, rounding to the nearest tenth.

Step-by-step explanation:

To find the measure of each angle in triangle ABC with vertices A(-1, 2), B(2, 8), and C(4, 1), we first need to calculate the lengths of the sides of the triangle using the distance formula. The length of side AB, BC, and CA are found as follows:

1. Length of AB: \(\sqrt{(2 - (-1))^2 + (8 - 2)^2} = \sqrt{9 + 36} = \sqrt{45}\)
2. Length of BC: \(\sqrt{(4 - 2)^2 + (1 - 8)^2} = \sqrt{4 + 49} = \sqrt{53}\)
3. Length of CA: \(\sqrt{((-1) - 4)^2 + (2 - 1)^2} = \sqrt{25 + 1} = \sqrt{26}\)

Next, we use the Law of Cosines to find the angles. For example, to find angle A:

\(\cos A = \frac{BC^2 + CA^2 - AB^2}{2 \cdot BC \cdot CA}\)

Substituting the lengths, we get:

\(\cos A = \frac{\sqrt{53}^2 + \sqrt{26}^2 - \sqrt{45}^2}{2 \cdot \sqrt{53} \cdot \sqrt{26}}\)

\(\cos A = \frac{53 + 26 - 45}{2 \cdot \sqrt{53} \cdot \sqrt{26}}\)

\(\cos A = \frac{34}{2 \cdot \sqrt{53} \cdot \sqrt{26}}\)

Now, we calculate the actual angle by taking the inverse cosine:

\(A = \cos^{-1}\left(\frac{34}{2 \cdot \sqrt{53} \cdot \sqrt{26}}\right)\)

We do the same for angles B and C. Finally, we round the answers to the nearest tenth to find the measure of each angle.