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3 problems match “right triangle trigonometry”. Clear

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1. Apothem of a regular pentagon ●●○○○ 20 ptsA regular pentagon has sides of length 4 m. Find the length of its apothem, in metres. Round to two decimal places.
2. Segment LM in a 30-60-90 triangle ●●●○○ 50 ptsIn triangle $ABC$, $\angle B=30^{\circ}$ and $\angle A=90^{\circ}$. On side $AC$, point $K$ is marked, and on side $BC$, points $L$ and $M$ are marked such that $KL=KM$ (point $L$ lies on segment $BM$). Given $AK=4$, $BL=31$, $MC=3$, find the length of segment $LM$.
3. Angle PCQ in a unit square ●●●○○ 50 ptsGiven a square $ABCD$ with side length $1$, points $P$ and $Q$ lie on sides $AB$ and $AD$, respectively. If the perimeter of $\triangle APQ$ is $2$, find the angle $\angle PCQ$ in degrees.