To determine the type of triangle formed by the points [tex]\( A (0, -8) \)[/tex], [tex]\( B (9, -2) \)[/tex], and [tex]\( C (-12, 10) \)[/tex], follow these steps:
1. Calculate the lengths of the sides:
To find the distance between two points, use the distance formula:
[tex]\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
2. Calculate the distance AB:
[tex]\[
AB = \sqrt{(9 - 0)^2 + (-2 - (-8))^2} = \sqrt{9^2 + 6^2} = \sqrt{81 + 36} = \sqrt{117} \approx 10.82
\][/tex]
3. Calculate the distance BC:
[tex]\[
BC = \sqrt{(-12 - 9)^2 + (10 - (-2))^2} = \sqrt{(-21)^2 + 12^2} = \sqrt{441 + 144} = \sqrt{585} \approx 24.19
\][/tex]
4. Calculate the distance CA:
[tex]\[
CA = \sqrt{(-12 - 0)^2 + (10 - (-8))^2} = \sqrt{(-12)^2 + 18^2} = \sqrt{144 + 324} = \sqrt{468} \approx 21.63
\][/tex]
5. List the sides and sort them:
[tex]\[
\text{Sides:} \quad AB \approx 10.82, \quad BC \approx 24.19, \quad CA \approx 21.63
\][/tex]
6. Determine the type of triangle:
- Check for Equilateral Triangle:
All three sides would need to be equal. Since [tex]\( AB \neq BC \neq CA \)[/tex], it is not equilateral.
- Check for Isosceles Triangle:
At least two sides need to be equal. Since [tex]\( AB \neq BC \neq CA \)[/tex], it is not isosceles.
- Check for Scalene Right Triangle:
For a right triangle, the Pythagorean theorem must hold true: [tex]\( a^2 + b^2 = c^2 \)[/tex] for the largest side [tex]\( c \)[/tex] (the hypotenuse).
[tex]\[
10.82^2 + 21.63^2 \approx 24.19^2
\][/tex]
[tex]\[
117 + 468 \approx 585 \quad \text{or} \quad 585 \approx 585
\][/tex]
This holds true, confirming it is a right triangle.
7. Conclusion:
Since it is a right triangle and all sides are of different lengths, it is a Scalene Right Triangle.
Result: The triangle formed by the points [tex]\( A(0, -8) \)[/tex], [tex]\( B(9, -2) \)[/tex], and [tex]\( C(-12, 10) \)[/tex] is a Scalene Right Triangle.
