Answer:
A) AB ⊥ AC
B) The triangle is a right triangle.
C) The triangle is an isosceles triangle
Step-by-step explanation:
we know that
the formula to calculate the distance between two points is equal to
[tex]d=\sqrt{(y2-y1)^{2}+(x2-x1)^{2}}[/tex]
step 1
Find the distance AB
we have
[tex]A(-1,3), B(-5,-1)[/tex]
substitute in the formula
[tex]d=\sqrt{(-1-3)^{2}+(-5+1)^{2}}[/tex]
[tex]d=\sqrt{(-4)^{2}+(-4)^{2}}[/tex]
[tex]d_A_B=\sqrt{32}\ units[/tex]
step 2
Find the distance BC
we have
[tex]B(-5,-1),C(3,-1)[/tex]
substitute in the formula
[tex]d=\sqrt{(-1+1)^{2}+(3+5)^{2}}[/tex]
[tex]d=\sqrt{(0)^{2}+(8)^{2}}[/tex]
[tex]d_B_C=8\ units[/tex]
step 3
Find the distance AC
we have
[tex]A(-1,3),C(3,-1)[/tex]
substitute in the formula
[tex]d=\sqrt{(-1-3)^{2}+(3+1)^{2}}[/tex]
[tex]d=\sqrt{(-4)^{2}+(4)^{2}}[/tex]
[tex]d_A_C=\sqrt{32}\ units[/tex]
step 4
Compare the length sides of triangle
[tex]d_A_B=\sqrt{32}\ units[/tex]
[tex]d_B_C=8\ units[/tex]
[tex]d_A_C=\sqrt{32}\ units[/tex]
therefore
The triangle ABC is an isosceles triangle, because has two equal sides
The triangle ABC is a right triangle because satisfy the Pythagoras theorem
[tex]BC^2=AB^2+AC^2[/tex]
[tex]8^2=(\sqrt{32})^2+(\sqrt{32})^2[/tex]
[tex]64=32+32[/tex]
[tex]64=64[/tex] ----> is true (Is a right triangle)
AB ⊥ AC because in a right triangle the legs are perpendicular
