Question

triangle abc is an isosceles triangle in which ab = ac. what is the perimeter of △abc? 15 units 10 + √10 units 3√10 units 5 + √10 units

Explanation:

Step1: Calculate the length of \(AB\)

Use the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
For points \(A(-1,6)\) and \(B(-1,1)\), \(x_1=-1,y_1 = 6,x_2=-1,y_2 = 1\).
\(AB=\sqrt{(-1+1)^2+(1 - 6)^2}=\sqrt{0+(-5)^2}=5\)

Step2: Calculate the length of \(AC\)

For points \(A(-1,6)\) and \(C(2,2)\), \(x_1=-1,y_1 = 6,x_2=2,y_2 = 2\).
\(AC=\sqrt{(2 + 1)^2+(2 - 6)^2}=\sqrt{9 + 16}=\sqrt{25}=5\)

Step3: Calculate the length of \(BC\)

For points \(B(-1,1)\) and \(C(2,2)\), \(x_1=-1,y_1 = 1,x_2=2,y_2 = 2\).
\(BC=\sqrt{(2 + 1)^2+(2 - 1)^2}=\sqrt{9+1}=\sqrt{10}\)

Step4: Calculate the perimeter of \(\triangle ABC\)

Perimeter \(P=AB + AC+BC\).
Since \(AB = 5\), \(AC = 5\) and \(BC=\sqrt{10}\), \(P=5 + 5+\sqrt{10}=10+\sqrt{10}\)

Answer:

\(10+\sqrt{10}\text{ units}\)