# 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 result:

d =  2.121 cm
e =  6.708 cm
f =  6.708 cm
E =  71.562 °
F =  71.562 °
D =  36.876 °

#### Solution:

$a=6 \ \text{cm} \ \\ d^2=(a/2)^2 + (a/2)^2 \ \\ \ \\ d=a/\sqrt{ 8 }=6/\sqrt{ 8 }=2.121 \ \text{cm}$
$e^2=a^2 + (a/2)^2=6^2 + (6/2)^2=45 \ \\ \ \\ e=\sqrt{ 1+1/4 } \cdot \ a=\sqrt{ 1+1/4 } \cdot \ 6=3 \ \sqrt{ 5 }=6.708 \ \text{cm}$
$f=e=6.7082=\dfrac{ 1677 }{ 250 }=6.708 \ \text{cm}$
$\angle FEB=45 \ ^\circ \ \\ \ \\ \sin AED=a/e \ \\ \ \\ AED=\dfrac{ 180^\circ }{ \pi } \cdot \arcsin(a/e)=\dfrac{ 180^\circ }{ \pi } \cdot \arcsin(6/6.7082) \doteq 63.4384 \ \\ \ \\ E=180 - AED - \angle FEB=180 - 63.4384 - 45=71.562 ^\circ =71^\circ 33'42"$
$F=E=71.5616=71.562 ^\circ =71^\circ 33'43"$
$D=180-F-E=180-71.562-71.5616=\dfrac{ 9219 }{ 250 }=36.876 ^\circ =36^\circ 52'34"$ Our examples were largely sent or created by pupils and students themselves. Therefore, we would be pleased if you could send us any errors you found, spelling mistakes, or rephasing the example. Thank you!

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