Question

nathan draws a triangle on the coordinate grid. he marks his points at (1,5), (4,1), and (-4,-5). what is the perimeter of the triangle, rounded to the nearest tenth of a unit? show all steps that nathan must take to find the solution. (4 points)

Explanation:

Step1: Use the distance formula

The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
Let \(A=(1,5)\), \(B=(4,1)\), \(C=(-4,-5)\)

For \(AB\):
\(x_1 = 1,y_1 = 5,x_2=4,y_2 = 1\)
\(d_{AB}=\sqrt{(4 - 1)^2+(1 - 5)^2}=\sqrt{3^2+(-4)^2}=\sqrt{9 + 16}=\sqrt{25}=5\)

Step2: Calculate \(BC\)

\(x_1 = 4,y_1 = 1,x_2=-4,y_2=-5\)
\(d_{BC}=\sqrt{(-4 - 4)^2+(-5 - 1)^2}=\sqrt{(-8)^2+(-6)^2}=\sqrt{64+36}=\sqrt{100}=10\)

Step3: Calculate \(AC\)

\(x_1 = 1,y_1 = 5,x_2=-4,y_2=-5\)
\(d_{AC}=\sqrt{(-4 - 1)^2+(-5 - 5)^2}=\sqrt{(-5)^2+(-10)^2}=\sqrt{25 + 100}=\sqrt{125}\approx11.2\)

Step4: Find the perimeter

Perimeter \(P=d_{AB}+d_{BC}+d_{AC}\)
\(P = 5+10 + 11.2=26.2\)

Answer:

The perimeter of the triangle is \(26.2\) units.