Sure! Let's determine whether the points [tex]\((-8, -1)\)[/tex], [tex]\((3, -8)\)[/tex], and [tex]\((3, -1)\)[/tex] form a right triangle. If they don't form a right triangle, we will determine whether the triangle is isosceles or scalene. I will walk you through each step to reach a conclusion.
### Step 1: Calculate the Squared Distances Between the Points
Firstly, we need to determine the squared distances between each pair of points. The formula for the distance between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ \text{Distance}^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2 \][/tex]
Let's calculate these squared distances.
1. Between Point 1 [tex]\((-8, -1)\)[/tex] and Point 2 [tex]\((3, -8)\)[/tex]:
[tex]\[ d_1 = (3 - (-8))^2 + (-8 - (-1))^2 \][/tex]
[tex]\[ d_1 = (3 + 8)^2 + (-8 + 1)^2 \][/tex]
[tex]\[ d_1 = 11^2 + (-7)^2 \][/tex]
[tex]\[ d_1 = 121 + 49 \][/tex]
[tex]\[ d_1 = 170 \][/tex]
2. Between Point 2 [tex]\((3, -8)\)[/tex] and Point 3 [tex]\((3, -1)\)[/tex]:
[tex]\[ d_2 = (3 - 3)^2 + (-1 - (-8))^2 \][/tex]
[tex]\[ d_2 = 0^2 + (-1 + 8)^2 \][/tex]
[tex]\[ d_2 = 0 + 7^2 \][/tex]
[tex]\[ d_2 = 49 \][/tex]
3. Between Point 3 [tex]\((3, -1)\)[/tex] and Point 1 [tex]\((-8, -1)\)[/tex]:
[tex]\[ d_3 = (-8 - 3)^2 + (-1 - (-1))^2 \][/tex]
[tex]\[ d_3 = (-8 - 3)^2 + 0^2 \][/tex]
[tex]\[ d_3 = (-11)^2 + 0 \][/tex]
[tex]\[ d_3 = 121 \][/tex]
### Step 2: Check for a Right Triangle
Now that we have the squared distances [tex]\(d_1\)[/tex], [tex]\(d_2\)[/tex], and [tex]\(d_3\)[/tex], we need to check if these distances satisfy the Pythagorean theorem. The theorem states that for a right triangle with sides [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] (where [tex]\(c\)[/tex] is the hypotenuse), the following must hold true:
[tex]\[ a^2 + b^2 = c^2 \][/tex]
We will check this by verifying if any combination of our squared distances equals another squared distance:
[tex]\[ d_1 + d_2 = 170 + 49 = 219 \quad \text{does not equal} \quad d_3 = 121 \][/tex]
[tex]\[ d_1 + d_3 = 170 + 121 = 291 \quad \text{does not equal} \quad d_2 = 49 \][/tex]
[tex]\[ d_2 + d_3 = 49 + 121 = 170 \quad \text{equals} \quad d_1 = 170 \][/tex]
Since one of the equations holds true, [tex]\(d_2 + d_3 = d_1\)[/tex], the points form a right triangle.
### Step 3: Conclusion
The points [tex]\((-8, -1)\)[/tex], [tex]\((3, -8)\)[/tex], and [tex]\((3, -1)\)[/tex] form a right triangle based on the squared distances satisfying the Pythagorean theorem.
So, the conclusive type of triangle formed by these points is a right triangle.
