Question

select the correct answer from each dropdown menu.
△abc has vertices of a(-2,5), b(-4,-2), and c(3,-4)
the length of ab is
the length of ac is
the length of bc is
therefore, the triangle is

Explanation:

Step1: Calculate the length of \(AB\)

Use the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
For \(A(-2,5)\) and \(B(-4,-2)\), \(x_1=-2,y_1 = 5,x_2=-4,y_2=-2\).
\(AB=\sqrt{(-4 + 2)^2+(-2 - 5)^2}=\sqrt{(-2)^2+(-7)^2}=\sqrt{4 + 49}=\sqrt{53}\)

Step2: Calculate the length of \(AC\)

For \(A(-2,5)\) and \(C(3,-4)\), \(x_1=-2,y_1 = 5,x_2=3,y_2=-4\).
\(AC=\sqrt{(3 + 2)^2+(-4 - 5)^2}=\sqrt{(5)^2+(-9)^2}=\sqrt{25+81}=\sqrt{106}\)

Step3: Calculate the length of \(BC\)

For \(B(-4,-2)\) and \(C(3,-4)\), \(x_1=-4,y_1=-2,x_2=3,y_2=-4\).
\(BC=\sqrt{(3 + 4)^2+(-4 + 2)^2}=\sqrt{(7)^2+(-2)^2}=\sqrt{49 + 4}=\sqrt{53}\)

Step4: Determine the type of triangle

Since \(AB=\sqrt{53}\), \(BC=\sqrt{53}\), two sides are equal.

Answer:

The length of \(AB\) is \(\text{square root of }53\). The length of \(AC\) is \(\text{square root of }106\). The length of \(BC\) is \(\text{square root of }53\). Therefore, the triangle is \(\text{isosceles}\).