Question

calculating the perimeter of a triangle. triangle abc is an isosceles triangle in which side ab = ac. what is the perimeter of triangle abc? o 5 + √10 units o 10 + √10 units o 10√10 units o 50 units

Explanation:

Step1: Find length of AC

Count the grid - units. A(-1,-4) and C(4,-4). Using the distance formula for points on a horizontal line (since y - coordinates are the same), the length of AC is \(|4 - (-1)|=5\) units.

Step2: Find length of AB

A(-1,-4) and B(2,-1). Use the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). Here \(x_1=-1,y_1 = - 4,x_2=2,y_2=-1\). Then \(AB=\sqrt{(2 - (-1))^2+(-1-(-4))^2}=\sqrt{(2 + 1)^2+(-1 + 4)^2}=\sqrt{3^2+3^2}=\sqrt{9 + 9}=\sqrt{18}=3\sqrt{2}\approx\sqrt{10}\) (approximate for the multiple - choice options). Since \(AB = AC\), \(AB=\sqrt{10}\) and \(AC = 5\), and \(BC\) can be found in the same way as \(AB\) and is also \(\sqrt{10}\).

Step3: Calculate perimeter

The perimeter \(P\) of \(\triangle ABC\) is \(P=AB + AC+BC\). Since \(AB = AC=\sqrt{10}\) and \(BC = 5\), \(P = 10+\sqrt{10}\) units.

Answer:

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