Given the vertices of a triangle at $(0,0)$, $(5,0)$, and $(2,4)$, use the Law of Cosines to find, correct to the nearest degree, the three angles of the triangle.

• A. $60°$, $60°$, $60°$
• B. $68°$, $90°$, $22°$
• C. $68°$, $56°$, $56°$
• D. $68°$, $80°$, $32°$

Answer: (D)

Verified step by step guidance

1. Label the vertices of the triangle as A(0,0), B(5,0), and C(2,4). To use the Law of Cosines, first find the lengths of the sides opposite each vertex by calculating the distance between the points: use the distance formula \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\) for each pair of points.
2. Calculate side lengths: \(a\) opposite vertex A is the length BC, \(b\) opposite vertex B is the length AC, and \(c\) opposite vertex C is the length AB. Substitute the coordinates into the distance formula to find \(a\), \(b\), and \(c\).
3. Apply the Law of Cosines to find each angle. For example, to find angle A, use the formula: \(\cos(A) = \frac{b^2 + c^2 - a^2}{2bc}\). Similarly, find angles B and C by cyclically permuting the sides in the formula.
4. Calculate the inverse cosine (arccos) of the values found in the previous step to get the angles in degrees. Remember to convert from radians to degrees if your calculator gives the result in radians.
5. Round each angle to the nearest degree. Verify that the sum of the three angles is approximately 180 degrees to confirm the correctness of your calculations.