Isosceles - figure

A figure consists of a dark square, two identical white isosceles triangles, and two identical white trapezoids. (With each side of the square coincides the base of one white figure.) The dark square has a side of length 12 cm and its area is half the area of the whole figure. One triangle has an area of 30 cm². The length of the shorter base of the trapezoid is 9 cm. Calculate in cm the height to the base of the isosceles triangle and the height of the trapezoid.

Final Answer:

h₁ =  5 cm
h₂ =  4 cm

Step-by-step explanation:

$$\begin{gathered} a=12\text{cm} \\ c=9\text{cm} \\ {S}_1=30{\text{cm}}^2 \\ {S}_2=a^2=12^2=144{\text{cm}}^2 \\ {S}_1=\frac{a\cdot h_1}{2} \\ {h}_1=\frac{2\cdot S_1}{a}=\frac{2\cdot 30}{12}=5\text{cm} \end{gathered}$$

$$\begin{gathered} S_2=2\cdot S_1+2\cdot S_3 \\ {S}_3=S_2/2-S_1=144/2-30=42{\text{cm}}^2 \\ {S}_3=\frac{a+c}{2}\cdot h_2 \\ {h}_2=\frac{2\cdot S_3}{a+c}=\frac{2\cdot 42}{12+9}\text{cm}=4\text{cm} \end{gathered}$$

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