MO Z6-I-3 2022

Magda cut out two identical isosceles triangles, each of which had a perimeter of 100 cm. First, from these triangles she formed a quadrilateral by placing them together by their legs. Then from them she formed a quadrilateral by placing them together by their bases. In the first case she obtained a quadrilateral with a perimeter 4 cm shorter than in the second case. Determine the lengths of the sides of the cut-out triangles.

Final Answer:

a =  34 cm
b =  32 cm
c =  34 cm

Step-by-step explanation:

$$\begin{gathered} o=100\text{cm} \\ a=c \\ o=2a+b \\ a+b+b+a=(2a+2a)-4 \\ 100=2a+b \\ 100=2\cdot a+b \\ a+b+b+a=(2\cdot a+2\cdot a)-4 \\ 2a+b=100 \\ 2a-2b=4 \\ Row2-Row1\to Row2 \\ -3b=-96 \\ b=\frac{-96}{-3}=32 \\ a=\frac{100-b}{2}=\frac{100-32}{2}=34 \\ a=34 \\ b=32 \end{gathered}$$

$c=a=34=34\text{cm}$

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