To the nearest tenth of a degree, find the angles of the triangle with vertices (0, 0), (6,3), and (1, 8). Use your protractor to check your calculations, and explain your method. DONT use some random complicated method or formula

To find the angles of a triangle with vertices \((0, 0)\), \((6, 3)\), and \((1, 8)\), we will follow these steps:

### 1. **Find the lengths of the sides:**
We can use the distance formula between points to determine the lengths of the sides of the triangle. The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

- **Side 1: Between \((0, 0)\) and \((6, 3)\)**
\[
\text{Length} = \sqrt{(6 - 0)^2 + (3 - 0)^2} = \sqrt{6^2 + 3^2} = \sqrt{36 + 9} = \sqrt{45} = 6.7
\]

- **Side 2: Between \((6, 3)\) and \((1, 8)\)**
\[
\text{Length} = \sqrt{(1 - 6)^2 + (8 - 3)^2} = \sqrt{(-5)^2 + (5)^2} = \sqrt{25 + 25} = \sqrt{50} = 7.1
\]

- **Side 3: Between \((0, 0)\) and \((1, 8)\)**
\[
\text{Length} = \sqrt{(1 - 0)^2 + (8 - 0)^2} = \sqrt{1^2 + 8^2} = \sqrt{1 + 64} = \sqrt{65} = 8.1
\]

Now we know the sides of the triangle are approximately 6.7, 7.1, and 8.1 units long.

### 2. **Use the Law of Cosines to find the angles:**
The Law of Cosines relates the sides of a triangle to the angles. For a triangle with sides \(a\), \(b\), and \(c\), and the angle \(C\) opposite side \(c\), the law states:
\[
\cos(C) = \frac{a^2 + b^2 - c^2}{2ab}
\]

We'll use this formula to find the angles.

- **Angle opposite the 8.1 unit side:**
\[
\cos(C) = \frac{6.7^2 + 7.1^2 - 8.1^2}{2 \cdot 6.7 \cdot 7.1} = \frac{44.89 + 50.41 - 65.61}{2 \cdot 6.7 \cdot 7.1} = \frac{29.69}{95.14} = 0.312
\]
\[
C = \cos^{-1}(0.312) \approx 72.1^\circ
\]

- **Angle opposite the 7.1 unit side:**
\[
\cos(B) = \frac{6.7^2 + 8.1^2 - 7.1^2}{2 \cdot 6.7 \cdot 8.1} = \frac{44.89 + 65.61 - 50.41}{2 \cdot 6.7 \cdot 8.1} = \frac{60.09}{108.54} = 0.554
\]
\[
B = \cos^{-1}(0.554) \approx 56.4^\circ
\]

- **Angle opposite the 6.7 unit side:**
Since the sum of the angles in a triangle is always \(180^\circ\), the remaining angle is:
\[
A = 180^\circ - 72.1^\circ - 56.4^\circ = 51.5^\circ
\]

### 3. **Summary of angles:**
The angles of the triangle are approximately:
- \( A \approx 51.5^\circ \)
- \( B \approx 56.4^\circ \)
- \( C \approx 72.1^\circ \)

### Protractor Check:
If you have a protractor, you can plot the triangle with the coordinates given, measure the angles directly at each vertex, and compare them to the calculated values to ensure they match closely.

Would you like more details or have any questions?

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Here are 5 related questions you could explore:
1. How does the Law of Cosines differ from the Law of Sines?
2. What happens if the triangle has an obtuse angle? How does that affect the calculations?
3. How can you confirm the triangle’s angle sum always adds up to 180°?
4. Is it possible to find the area of this triangle using these side lengths?
5. What are some practical applications of the Law of Cosines?

**Tip:** Always remember to check if a triangle is right-angled before using the Law of Cosines—it simplifies to the Pythagorean Theorem in such cases!

Learn how to find the angles of a triangle with vertices (0, 0), (6, 3), and (1, 8) using the distance formula and the Law of Cosines. This step-by-step guide is ideal for high school students and covers essential geometry and trigonometry concepts.

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