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:
a yes, because ab = bc = ac.
b yes, because (overline{ab}perpoverline{ac}).
c no, because (overline{bc}) is longer than (overline{ab}).
d no, because (overline{bc}) is not perpendicular to (overline{ab}).

Explanation:

Step1: Calculate length of AB

Use distance formula $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. For points $A(-4,6)$ and $B(6,6)$, $x_1=-4,y_1 = 6,x_2=6,y_2 = 6$. Then $AB=\sqrt{(6 - (-4))^2+(6 - 6)^2}=\sqrt{10^2+0^2}=10$.

Step2: Calculate length of BC

For points $B(6,6)$ and $C(1,-3)$, $x_1 = 6,y_1=6,x_2=1,y_2=-3$. Then $BC=\sqrt{(1 - 6)^2+(-3 - 6)^2}=\sqrt{(-5)^2+(-9)^2}=\sqrt{25 + 81}=\sqrt{106}$.

Step3: Analyze the results

Since $AB = 10$ and $BC=\sqrt{106}$, $AB
eq BC$. An equilateral triangle has all - sides equal. So triangle $ABC$ is not an equilateral triangle.

Answer:

C. No, because $\overline{BC}$ is longer than $\overline{AB}$.