To determine the type of triangle formed by the points [tex]\( D(7, 3) \)[/tex], [tex]\( E(8, 1) \)[/tex], and [tex]\( F(4, -1) \)[/tex], we need to follow these steps:
1. Calculate the distances between each pair of points:
- Distance between [tex]\( D \)[/tex] and [tex]\( E \)[/tex]:
[tex]\[
DE = \sqrt{(8 - 7)^2 + (1 - 3)^2} = \sqrt{1^2 + (-2)^2} = \sqrt{1 + 4} = \sqrt{5} \approx 2.236
\][/tex]
- Distance between [tex]\( E \)[/tex] and [tex]\( F \)[/tex]:
[tex]\[
EF = \sqrt{(4 - 8)^2 + (-1 - 1)^2} = \sqrt{(-4)^2 + (-2)^2} = \sqrt{16 + 4} = \sqrt{20} \approx 4.472
\][/tex]
- Distance between [tex]\( F \)[/tex] and [tex]\( D \)[/tex]:
[tex]\[
FD = \sqrt{(4 - 7)^2 + (-1 - 3)^2} = \sqrt{(-3)^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5.0
\][/tex]
2. Identify the lengths of the sides of the triangle:
[tex]\[
DE \approx 2.236, \quad EF \approx 4.472, \quad FD = 5.0
\][/tex]
3. Determine the type of triangle:
- Check if two sides are equal:
[tex]\[
DE \neq EF, \quad EF \neq FD, \quad FD \neq DE (\text{None of the sides are equal, so it is not an isosceles triangle})
\][/tex]
- Check if it is a right triangle:
Using the Pythagorean theorem, we check whether:
[tex]\[
(DE)^2 + (EF)^2 \approx (FD)^2
\][/tex]
Substituting the approximate values:
[tex]\[
(2.236)^2 + (4.472)^2 \approx 5^2
\][/tex]
[tex]\[
5 + 20 \approx 25
\][/tex]
[tex]\[
25 \approx 25 \quad (\text{This is true, hence the triangle is a right triangle})
\][/tex]
4. Conclude the type of triangle:
Based on the calculations, the triangle formed by the points [tex]\( D(7, 3) \)[/tex], [tex]\( E(8, 1) \)[/tex], and [tex]\( F(4, -1) \)[/tex] is a right triangle.
Therefore, the correct answer is:
C. right triangle
