Certainly! Let's solve the problem step by step.
Part A: Find the length of each side of the triangle
We'll use the distance formula to find the length of each side of the triangle. The distance formula between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
1. Length of side AB:
[tex]\[
A(0, 2), B(2, 5) \\
\text{AB} = \sqrt{(2 - 0)^2 + (5 - 2)^2} \\
\text{AB} = \sqrt{2^2 + 3^2} \\
\text{AB} = \sqrt{4 + 9} \\
\text{AB} = \sqrt{13} ≈ 3.605551275463989
\][/tex]
2. Length of side BC:
[tex]\[
B(2, 5), C(-1, 7) \\
\text{BC} = \sqrt{(-1 - 2)^2 + (7 - 5)^2} \\
\text{BC} = \sqrt{(-3)^2 + 2^2} \\
\text{BC} = \sqrt{9 + 4} \\
\text{BC} = \sqrt{13} ≈ 3.605551275463989
\][/tex]
3. Length of side CA:
[tex]\[
C(-1, 7), A(0, 2) \\
\text{CA} = \sqrt{(0 - (-1))^2 + (2 - 7)^2} \\
\text{CA} = \sqrt{1^2 + (-5)^2} \\
\text{CA} = \sqrt{1 + 25} \\
\text{CA} = \sqrt{26} ≈ 5.0990195135927845
\][/tex]
Summary of the lengths:
[tex]\[
AB ≈ 3.605551275463989 \\
BC ≈ 3.605551275463989 \\
CA ≈ 5.0990195135927845
\][/tex]
Part B: Find the slope of each side of the triangle
The slope [tex]\( m \)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
1. Slope of side AB:
[tex]\[
A(0, 2), B(2, 5) \\
\text{Slope of AB} = \frac{5 - 2}{2 - 0} = \frac{3}{2} = 1.5
\][/tex]
2. Slope of side BC:
[tex]\[
B(2, 5), C(-1, 7) \\
\text{Slope of BC} = \frac{7 - 5}{-1 - 2} = \frac{2}{-3} = -\frac{2}{3} ≈ -0.6666666666666666
\][/tex]
3. Slope of side CA:
[tex]\[
C(-1, 7), A(0, 2) \\
\text{Slope of CA} = \frac{2 - 7}{0 - (-1)} = \frac{-5}{1} = -5.0
\][/tex]
Summary of the slopes:
[tex]\[
\text{Slope of AB} = 1.5 \\
\text{Slope of BC} ≈ -0.6666666666666666 \\
\text{Slope of CA} = -5.0
\][/tex]
Part C: Classify the triangle
A triangle is classified based on the lengths of its sides:
- An equilateral triangle has all three sides equal.
- An isosceles triangle has at least two sides equal.
- A scalene triangle has all sides of different lengths.
Length of sides:
[tex]\[
AB ≈ 3.605551275463989 \\
BC ≈ 3.605551275463989 \\
CA ≈ 5.0990195135927845
\][/tex]
Since sides AB and BC are equal and CA is different, the triangle is classified as isosceles.
Classification:
[tex]\[
\text{Triangle ABC is isosceles because two sides (AB and BC) are of equal length.}
\][/tex]
And with that, we have completed all parts of the question!
