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Macedonian Mathematical Competitions

Geometry Difficulty 5.5 AIME, harder Prove it North Macedonia

In the isosceles triangle $ABC$ let $D$ be the base point for the altitude from $C$ to the base $AB$ of $ABC$ . Calculate the length of the segment that connects the middle point of $AD$ and the middle point of $AC$ if it is known that the perimeter of the triangle $ABC$ is $36\, ext{cm}$ and the perimeter of the triangle $ADC$ is $29\, ext{cm}$ .

Solution

$2 \cdot L_{\Delta ADC} = L_{\Delta ABC} + 2 \cdot \overline{CD}$ . From here, we obtain $2 \cdot 29 = 36 + 2 \cdot \overline{CD}$ or $\overline{CD} = (2 \cdot 29 - 36) : 2 = 11\,\text{cm}$ . Let $M$ be the middle point of $AD$ and $N$ be the middle point of $AC$ . Then $MN$ is the middle line of the triangle $ADC$ so we have $\overline{MN} = \frac{1}{2} \overline{CD} = \frac{1}{2} \cdot 11 = 5.5\,\text{cm}$ .

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