Question

jenna and mike are given two points of a triangle, a(-2,-4) and b(8,6).
in order to create an isosceles triangle, where ab = ac, jenna says point c must be (-10,7) and mike says point c must be (12,-6)
which statement is true?
a. neither jenna nor mike are correct.
b. both jenna and mike are correct.
c. mike is correct.
d. jenna is correct.

Explanation:

Step1: Calculate the distance \(AB\)

Use the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
For \(A(-2,-4)\) and \(B(8,6)\), \(x_1=-2,y_1 = - 4,x_2=8,y_2 = 6\).
\(AB=\sqrt{(8+2)^2+(6 + 4)^2}=\sqrt{100 + 100}=\sqrt{200}=10\sqrt{2}\)

Step2: Calculate the distance \(AC\) for Jenna's point \(C(-10,7)\)

For \(A(-2,-4)\) and \(C(-10,7)\), \(x_1=-2,y_1=-4,x_2=-10,y_2 = 7\)
\(AC=\sqrt{(-10 + 2)^2+(7 + 4)^2}=\sqrt{64+121}=\sqrt{185}
eq10\sqrt{2}\)

Step3: Calculate the distance \(AC\) for Mike's point \(C(12,-6)\)

For \(A(-2,-4)\) and \(C(12,-6)\), \(x_1=-2,y_1=-4,x_2=12,y_2=-6\)
\(AC=\sqrt{(12 + 2)^2+(-6 + 4)^2}=\sqrt{196 + 4}=\sqrt{200}=10\sqrt{2}\)

Answer:

C. Mike is correct.