In Exercises 35–36, the three given points are the vertices of a triangle. Solve each triangle, rounding lengths of sides to the nearest tenth and angle measures to the nearest degree.
A(0, 0), B(-3, 4), C(3, -1)

Verified step by step guidance

1. Identify the vertices of the triangle as points A(0, 0), B(-3, 4), and C(3, -1). The goal is to find all side lengths and angle measures of triangle ABC.
2. Calculate the lengths of the sides using the distance formula between two points: \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). Specifically, find \(AB\), \(BC\), and \(AC\) by substituting the coordinates of the points.
3. Once the side lengths are found, use the Law of Cosines to find each angle. For example, to find angle \(A\), use the formula: \(\cos A = \frac{b^2 + c^2 - a^2}{2bc}\), where \(a\), \(b\), and \(c\) are the lengths of the sides opposite to angles \(A\), \(B\), and \(C\) respectively.
4. Calculate the cosine values for each angle using the Law of Cosines, then find the angle measures by taking the inverse cosine (arccos) of those values.
5. Round the side lengths to the nearest tenth and the angle measures to the nearest degree to complete the solution of the triangle.

Key Concepts

Distance Formula

The distance formula calculates the length between two points in the coordinate plane using their coordinates. It is derived from the Pythagorean theorem and given by √((x2 - x1)² + (y2 - y1)²). This formula is essential for finding the side lengths of the triangle formed by the given vertices.

Law of Cosines

The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. It is useful for finding unknown angles or sides when two sides and the included angle or all three sides are known. This law helps solve the triangle once side lengths are determined.

Triangle Angle Sum Property

The triangle angle sum property states that the sum of the interior angles of any triangle is always 180 degrees. After calculating two angles using the Law of Cosines, this property allows finding the third angle by subtracting the sum of the known angles from 180 degrees.