Question

Triangle ABC has the following vertices:A(-4,6)B(6,6)C(1,-3)Is triangle ABC an equilateral triangle, and why?Choose 1 answer:Yes, because AB = BC=AC.Yes, because AB 1 AC.No, because BC is longer than AB.No, because BC is not perpendicular to AB.

Answer 1

Explanation

We are given the following:

$\begin{gathered} A(-4,6) \\ B(6,6) \\ C(1,-3) \end{gathered}$

We are required to determine whether or not triangle ABC is an equilateral triangle.

We know that an equilateral triangle is a triangle with all sides equal.

We also know that the distance between two points is given as:

$Distance=\sqrt{(y_2-y_1)^2+(x_2-x_1)^2}$

- The distance AB becomes:

$\begin{gathered} A(-4,6)\to(x_1,y_1) \\ B(6,6)\to(x_2,y_2) \\ AB=\sqrt{(y_2-y_1)^2+(x_2-x_1)^2} \\ AB=\sqrt{(6-6)^2+(6-(-4))^2} \\ AB=\sqrt{0^2+10^2}=\sqrt{0+100}=\sqrt{100} \\ AB=10\text{ units } \end{gathered}$

- The distance BC becomes:

$\begin{gathered} B(6,6)\to(x_1,y_1) \\ C(1,-3)\to(x_2,y_2) \\ BC=\sqrt{(y_2-y_1)^2+(x_2-x_1)^2} \\ BC=\sqrt{(-3-6)^2+(1-6)^2} \\ BC=\sqrt{(-9)^2+(-5)^2}=\sqrt{81+25}=\sqrt{106} \\ BC\approx10.3\text{ units} \end{gathered}$

Hence, the answer is:

$\text{ No, because BC is longer than AB}$

Option C is correct.