Triangle in a square

In a square ABCD with side a = 6 cm, point E is the center of side AB, and point F is the center of side BC. Calculate the size of all angles of the triangle DEF and the lengths of its sides.

Correct answer:

d =  2.1213 cm
e =  6.7082 cm
f =  6.7082 cm
E =  71.5651 °
F =  71.5651 °
D =  36.8698 °

Step-by-step explanation:

$$\begin{gathered} a=6\text{cm} \\ {d}^2=(a/2{)}^2+(a/2{)}^2 \\ d=a/\sqrt{8}=6/\sqrt{8}=2.1213\text{cm} \end{gathered}$$

$$\begin{gathered} e^2=a^2+(a/2{)}^2=6^2+(6/2{)}^2=45 \\ e=\sqrt{1+1/4}\cdot a=\sqrt{1+1/4}\cdot 6=3\sqrt{5}=6.7082\text{cm} \end{gathered}$$

$f=e=6.7082=3\sqrt{5}=6.7082\text{cm}$

$$\begin{gathered} \angle FEB=45{}^{\circ } \\ \sin AED=a/e \\ AED=\frac{180°}{\pi }\cdot \arcsin (a/e)=\frac{180°}{\pi }\cdot \arcsin (6/6.7082)\doteq 63.4349 \\ E=180-\angle AED-\angle FEB=180-63.4349-45=71.5651^{\circ }=71°33^{\prime }54" \end{gathered}$$

$F=E=71.5651=71.5651^{\circ }=71°33^{\prime }54"$

$D=180-F-E=180-71.5651-71.5651=36.8698^{\circ }=36°52^{\prime }11"$

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