Question

what is the perimeter of stu? write your answer as an integer or as a decimal rounded to the nearest tenth. perimeter = units. find the tangent of ∠f. simplify your answer and write it as a proper fraction, improper fraction, or whole number. tan(f) =

Explanation:

Step1: Find side - lengths in the first triangle

For $\triangle STU$:

• Using the distance formula $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. Let $S(- 4,0)$, $T(2,0)$ and $U(-1,3)$.
• $ST=\vert-4 - 2\vert=6$.
• $SU=\sqrt{(-1 + 4)^2+(3 - 0)^2}=\sqrt{9 + 9}=\sqrt{18}=3\sqrt{2}$.
• $TU=\sqrt{(-1 - 2)^2+(3 - 0)^2}=\sqrt{9 + 9}=\sqrt{18}=3\sqrt{2}$.

Step2: Calculate the perimeter of $\triangle STU$

Perimeter $P=ST + SU+TU=6 + 3\sqrt{2}+3\sqrt{2}=6 + 6\sqrt{2}\approx6+6\times1.414 = 6+8.484\approx14.5$.

Step3: Find the tangent in the second triangle

In right - triangle $DEF$ with right - angle at $E$, $\tan(F)=\frac{DE}{EF}$. Given $DE = 12$ and $EF = 5$.
So, $\tan(F)=\frac{12}{5}$.

Answer:

Perimeter $\approx14.5$ units
$\tan(F)=\frac{12}{5}$