To determine the type of triangle [tex]\(\Delta EFG\)[/tex] formed by the points [tex]\(E(0,0)\)[/tex], [tex]\(F(-7,4)\)[/tex], and [tex]\(G(0,8)\)[/tex], we need to calculate the lengths of the sides and examine their properties.
1. Calculating the lengths of the sides:
- The length [tex]\(EF\)[/tex] between points [tex]\(E(0,0)\)[/tex] and [tex]\(F(-7,4)\)[/tex]:
[tex]\[
EF = \sqrt{(-7 - 0)^2 + (4 - 0)^2} = \sqrt{49 + 16} = \sqrt{65} \approx 8.062
\][/tex]
- The length [tex]\(FG\)[/tex] between points [tex]\(F(-7,4)\)[/tex] and [tex]\(G(0,8)\)[/tex]:
[tex]\[
FG = \sqrt{(0 - (-7))^2 + (8 - 4)^2} = \sqrt{49 + 16} = \sqrt{65} \approx 8.062
\][/tex]
- The length [tex]\(GE\)[/tex] between points [tex]\(G(0,8)\)[/tex] and [tex]\(E(0,0)\)[/tex]:
[tex]\[
GE = \sqrt{(0 - 0)^2 + (8 - 0)^2} = \sqrt{0 + 64} = \sqrt{64} = 8.0
\][/tex]
2. Determining the properties of the triangle:
- To check if the triangle is scalene (all sides have different lengths):
[tex]\[
\text{EF} \approx 8.062 \neq \text{FG} \approx 8.062 \text{ and } \text{FG} \approx 8.062 \neq \text{GE} = 8.0 \text{ and } \text{GE} = 8.0 \neq \text{EF} \approx 8.062
\][/tex]
Since at least two sides are equal, the triangle is not scalene.
- To check if the triangle is isosceles (at least two sides have the same length):
[tex]\[
\text{EF} \approx 8.062 = \text{FG} \approx 8.062
\][/tex]
Since two of its sides are equal, the triangle is isosceles.
- To check if the triangle is equilateral (all three sides have the same length):
[tex]\[
\text{EF} \approx 8.062 \neq \text{GE} = 8.0
\][/tex]
Since not all three sides are equal, the triangle is not equilateral.
- To check if the triangle is a right triangle (Pythagorean theorem):
[tex]\[
\text{EF}^2 + \text{FG}^2 \approx (8.062)^2 + (8.062)^2 = 65 + 65 = 130 \neq (8.0)^2 = 64
\][/tex]
The side lengths do not satisfy the Pythagorean theorem, so the triangle is not a right triangle.
Since the triangle is not scalene, equilateral, or right, the correct statement is:
- [tex]\(\triangle EFG\)[/tex] is an isosceles triangle.
So, the correct choice is:
[tex]\[
\boxed{\triangle EFG \text{ is an isosceles triangle.}}
\][/tex]
