goofy question

[half shell]

An equilateral triangle is inscribed in a square as shown in the diagram. If the triangle has equal sides of length 1, what is the length of the side of the square? I am having trouble forming the equations for the simultaneous because I find no solution with my working out, I would appreciate it if you help me with telling me the first two equations I can form from this.

[fierce light]

try finding the angle USR, then use trig to find the length of SR

[opaque portal]

You can find the side length $s$ by saying the area of the square, $s^2$ is the same as the sum of the areas of the four triangles. Doing this, I get $s^2 =\frac{1+\sqrt{3}}{2}$

[river yacht]

Jay

[tropic wyvern]

PT=UR=a
2LΔ=2×1/2×s×a=sa
LΔTQU =1/2×(s-a)²

tΔTSU=(1²-(1/2)²)½=(3/4)½
LΔTSU=1/2×(3/4)½

Luas persegi s²=2LΔ+LΔTQU+LΔTSU
s²=sa+1/2(s-a)²+1/2(3/4)½
s²=sa+1/2(s²+a²-2sa)+1/2(3/4)½
s²=sa+1/2s²+1/2a²-sa+1/2(3/4)½
1/2s²=1/2a²+1/2(3/4)½
1/2s²=1/2(a²+(3/4)½)
s²=a²+(3/4)½
s=[a²+(3/4)½]½

[tropic wyvern]

Another way to find out

since ΔTUQ has 2 same length of side=s-a, then <TUQ=45º
<TUR=180-45=135º
<SUR=135-60⁰=75º
<USR=180-75-90=15º
s/1=cos15º
s/1=cos45.cos30+sin45.sin30
s=(2)½/2.(3)½/2+((2)½/2.1/2)
s=(6)½/4+(2)½/4
s=(6½+2½)/4

[half shell]

ty

.close