Question

question 6(multiple choice worth 1 points) (04.01 mc) △abc is located at a(3, 1), b(4, 4), and c(5, 1). zackery says that △abc is an equilateral triangle, while verna says that it is an isosceles triangle. who is correct? o zackery, because all of the sides are congruent o zackery, because there are no congruent sides o verna, because bc ≅ ac o verna, because ab ≅ bc

Explanation:

Step1: Calculate distance between points

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. For points $A(3,1)$ and $B(4,4)$:
\[AB=\sqrt{(4 - 3)^2+(4 - 1)^2}=\sqrt{1 + 9}=\sqrt{10}\]

Step2: Calculate another side - length

For points $B(4,4)$ and $C(5,1)$:
\[BC=\sqrt{(5 - 4)^2+(1 - 4)^2}=\sqrt{1+9}=\sqrt{10}\]

Step3: Calculate the third side - length

For points $A(3,1)$ and $C(5,1)$:
\[AC=\sqrt{(5 - 3)^2+(1 - 1)^2}=\sqrt{4+0}=2\]
Since $AB = BC=\sqrt{10}$ and $AB
eq AC$, $BC
eq AC$, the triangle is isosceles.

Answer:

Verna, because $AB = BC$