Sure! Let's determine the various lengths and the type of triangle step by step:
1. Finding the Length of [tex]\(AB\)[/tex]:
Using the distance formula [tex]\( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)[/tex], where [tex]\(A (-2,5)\)[/tex] and [tex]\(B(-4,-2)\)[/tex]:
[tex]\[
AB = 7.280109889280518
\][/tex]
2. Finding the Length of [tex]\(AC\)[/tex]:
Using the distance formula again, where [tex]\(A (-2,5)\)[/tex] and [tex]\(C (3,-4)\)[/tex]:
[tex]\[
AC = 10.295630140987
\][/tex]
3. Finding the Length of [tex]\(BC\)[/tex]:
Using the distance formula again, where [tex]\(B(-4,-2)\)[/tex] and [tex]\(C (3,-4)\)[/tex]:
[tex]\[
BC = 7.280109889280518
\][/tex]
4. Determining the Type of Triangle:
To determine the type of triangle, we compare the lengths of its sides:
- [tex]\(AB = 7.280109889280518\)[/tex]
- [tex]\(AC = 10.295630140987\)[/tex]
- [tex]\(BC = 7.280109889280518\)[/tex]
Since [tex]\(AB = BC\)[/tex] and [tex]\(AC \neq AB \neq BC\)[/tex], two sides are equal, and one side is different. Therefore, the triangle is isosceles.
So the complete answer is:
[tex]\[
\text{The length of } AB \text{ is } 7.280109889280518 \text{ The length of } AC \text{ is } 10.295630140987 \text{ The length of } BC \text{ is } 7.280109889280518 \text{ Therefore, the triangle is isosceles}
\][/tex]
