Discover the best answers at Westonci.ca, where experts share their insights and knowledge with you. Get the answers you need quickly and accurately from a dedicated community of experts on our Q&A platform. Get quick and reliable solutions to your questions from a community of experienced experts on our platform.
Sagot :
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.
### 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.
Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Thank you for visiting Westonci.ca. Stay informed by coming back for more detailed answers.
