Question

find the perimeter of the triangle with these vertices. (-4,5),(6,2),(-4,2) give an exact answer (not a decimal approximation). simplify your answer as much as possible. scratch area (not part of answer) perimeter:

Explanation:

Step1: Find the length of the vertical side

The two points \((-4,5)\) and \((-4,2)\) have the same \(x -\)coordinate. Use the distance formula for vertical points \(d=\vert y_2 - y_1\vert\).
\(d_1=\vert5 - 2\vert=3\)

Step2: Find the length of the horizontal side

The two points \((6,2)\) and \((-4,2)\) have the same \(y -\)coordinate. Use the distance formula for horizontal points \(d=\vert x_2 - x_1\vert\).
\(d_2=\vert6-(-4)\vert=\vert6 + 4\vert = 10\)

Step3: Find the length of the hypotenuse

Use the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) for the points \((-4,5)\) and \((6,2)\).
\(d_3=\sqrt{(6-(-4))^2+(2 - 5)^2}=\sqrt{(6 + 4)^2+(-3)^2}=\sqrt{100 + 9}=\sqrt{109}\)

Step4: Calculate the perimeter

The perimeter \(P\) of a triangle is \(P=d_1 + d_2 + d_3\).
\(P=3+10+\sqrt{109}=13+\sqrt{109}\)

Answer:

\(13+\sqrt{109}\)