Each letter represents one side of a right triangle.
We can find the length of side c as follows:
[tex]\begin{gathered} \sin (60\degree)=\frac{1.5}{c} \\ c=\frac{1.5}{\sin(60\degree)} \\ c=\frac{1.5}{\frac{\sqrt[]{3}}{2}} \\ c\approx1.732\text{ }cm \end{gathered}[/tex]
Sides c and d are congrent, then:
[tex]d\approx1.732\text{ }cm[/tex]
Applying the Pythagorean theorem to the triangle formed by sides c = 1.732, h, and the side of 1.5 cm, we get:
[tex]\begin{gathered} 1.732^2=1.5^2+h^2 \\ 3=2.25+h^2 \\ 3-2.25=h^2 \\ \sqrt[]{0.75}=h \\ 0.866\text{ }cm\approx h \end{gathered}[/tex]
Applying the Pythagorean theorem to the triangle formed by sides 1.5 cm, d, and i, the same result is got, then:
[tex]i\approx0.866\text{ }cm[/tex]
We can find the length of side e as follows:
[tex]\begin{gathered} \cos (55\degree)=\frac{h+i}{e} \\ \cos (55\degree)=\frac{0.866+0.866}{e} \\ e=\frac{0.866+0.866}{\cos (55\degree)} \\ e\approx3.02\text{ }cm \end{gathered}[/tex]
Applying the Pythagorean theorem, we can find the length of side j as follows:
[tex]\begin{gathered} e^2=j^2+(h+i)^2 \\ 3.02^2=j^2+(0.866+0.866)^2 \\ 9.1204=j^2+3 \\ 9.1204-3=j^2 \\ \sqrt[]{6.1204}=j \\ j\approx2.474\text{ }cm \end{gathered}[/tex]
We can find the length of side b as follows:
[tex]\begin{gathered} \sin (55\degree)=\frac{b}{j} \\ b=\sin (55\degree)\cdot2.474 \\ b\approx2.026\text{ }cm \end{gathered}[/tex]
We can find the length of side l as follows:
[tex]\begin{gathered} \cos (55\degree)=\frac{l}{2.474} \\ l=\cos (55\degree)\cdot2.474 \\ l\approx1.42\text{ }cm \end{gathered}[/tex]
Considering that sides k and j are congruent, if we would apply the Pythagorean theorem to triangles with sides b, l, j, and a, l, k, we will conclude that sides a and b are congruent, then:
[tex]a\approx2.026\text{ }cm[/tex]
Sides a, f, g, l form a rectangle, then f and a are congruent, and l and g are also congruent, that is,
[tex]\begin{gathered} f\approx2.026\text{ }cm \\ g\approx1.42\text{ }cm \end{gathered}[/tex]
Finally, the perimeter of the figure shown is:
[tex]\begin{gathered} P=a+b+c+d+e+f+g \\ P\approx2.026+2.026+1.732+1.732+3.02+2.026+1.42 \\ P\approx14\text{ }cm \end{gathered}[/tex]
