Question

what is the perimeter of △stu? perimeter = units

Explanation:

Step1: Determine side - lengths using grid

Assume each grid square has side - length of 1 unit. Let's assume the coordinates of \(S(-9,7)\), \(T(-9,10)\), \(U(-6,10)\).
The length of \(ST\) is the difference in \(y\) - coordinates since \(x\) - coordinates are the same. \(ST=\vert10 - 7\vert=3\) units.
The length of \(TU\) is the difference in \(x\) - coordinates since \(y\) - coordinates are the same. \(TU=\vert-6-(-9)\vert = 3\) units.

Step2: Calculate length of \(SU\) using Pythagorean theorem

The change in \(x\) from \(S\) to \(U\) is \(\Delta x=-6-(-9) = 3\) and the change in \(y\) is \(\Delta y=10 - 7=3\).
By the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(a = 3\), \(b = 3\), then \(SU=\sqrt{3^{2}+3^{2}}=\sqrt{9 + 9}=\sqrt{18}=3\sqrt{2}\) units.

Step3: Calculate the perimeter

The perimeter \(P\) of \(\triangle STU\) is \(P=ST + TU+SU\).
\(P=3 + 3+3\sqrt{2}=6 + 3\sqrt{2}\approx6+3\times1.414=6 + 4.242 = 10.242\approx10.24\) units.

Answer:

\(6 + 3\sqrt{2}\approx10.24\)