To solve the problem, we will go through each part step by step.
### Part A: Finding the Length of Each Side of the Triangle (4 points)
To find the length of each side of the triangle, we use the distance formula:
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
1. Length of AB:
[tex]\[ A = (0, 2), B = (2, 5) \][/tex]
[tex]\[ \text{AB length} = \sqrt{(2 - 0)^2 + (5 - 2)^2} = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13} \approx 3.6056 \][/tex]
2. Length of BC:
[tex]\[ B = (2, 5), C = (-1, 7) \][/tex]
[tex]\[ \text{BC length} = \sqrt{(-1 - 2)^2 + (7 - 5)^2} = \sqrt{(-3)^2 + 2^2} = \sqrt{9 + 4} = \sqrt{13} \approx 3.6056 \][/tex]
3. Length of AC:
[tex]\[ A = (0, 2), C = (-1, 7) \][/tex]
[tex]\[ \text{AC length} = \sqrt{(-1 - 0)^2 + (7 - 2)^2} = \sqrt{(-1)^2 + 5^2} = \sqrt{1 + 25} = \sqrt{26} \approx 5.0990 \][/tex]
So, the lengths of the sides are:
- AB: 3.6056
- BC: 3.6056
- AC: 5.0990
### Part B: Finding the Slope of Each Side of the Triangle (3 points)
To find the slope of each side of the triangle, we use the slope formula:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
1. Slope of AB:
[tex]\[ A = (0, 2), B = (2, 5) \][/tex]
[tex]\[ \text{AB slope} = \frac{5 - 2}{2 - 0} = \frac{3}{2} = 1.5 \][/tex]
2. Slope of BC:
[tex]\[ B = (2, 5), C = (-1, 7) \][/tex]
[tex]\[ \text{BC slope} = \frac{7 - 5}{-1 - 2} = \frac{2}{-3} = -\frac{2}{3} \approx -0.6667 \][/tex]
3. Slope of AC:
[tex]\[ A = (0, 2), C = (-1, 7) \][/tex]
[tex]\[ \text{AC slope} = \frac{7 - 2}{-1 - 0} = \frac{5}{-1} = -5.0 \][/tex]
So, the slopes of the sides are:
- AB: 1.5
- BC: -0.6667
- AC: -5.0
### Part C: Classifying the Triangle (3 points)
We classify the triangle based on the lengths of its sides:
1. Equilateral Triangle: All three sides are the same length.
2. Isosceles Triangle: Exactly two sides are the same length.
3. Scalene Triangle: All three sides are different lengths.
Given the side lengths:
- AB: 3.6056
- BC: 3.6056
- AC: 5.0990
We observe that the lengths of AB and BC are equal, while AC is different.
Therefore, the triangle is an Isosceles Triangle because it has exactly two sides of equal length.
### Summary:
- Side Lengths:
- AB ≈ 3.6056
- BC ≈ 3.6056
- AC ≈ 5.0990
- Slopes:
- AB: 1.5
- BC: -0.6667
- AC: -5.0
- Classification: Isosceles Triangle
This completes the detailed solution for the given problem.
