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30-60-90 triangles

Solve the first problem 10 problems

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1. Third side of a 30-60-90 triangle ●○○○○ 10 ptsA right triangle has two interior angles of $30^\circ$ and $90^\circ$, and two of its side lengths are $5$ and $5\sqrt{3}$. Find the third side length.
2. Sum of legs in a 30-60-90 triangle ●○○○○ 10 ptsA 30-60-90 right triangle has hypotenuse $2\sqrt{6}$ centimeters. Find the sum of the lengths, in centimeters, of its two legs. Round to two decimal places.
3. Sides of a 30-60-90 triangle ●○○○○ 10 ptsTriangle $ABC$ has angles $30^\circ$, $60^\circ$ and $90^\circ$, and its shortest side has length $3$ units. Find the lengths of the other two sides, in increasing order. Round to two decimal places.
4. Hypotenuse of a 30-60-90 triangle ●●○○○ 20 ptsIn right triangle $ABD$, the right angle is at $D$ and $\angle ABD = 60^\circ$. The inscribed circle of the triangle has radius 6 cm. Find the length of $\overline{AB}$ in centimeters. Round to two decimal places.
5. Area of a 30-60-90 triangle ●●○○○ 20 ptsA right triangle has angles of $30^\circ$ and $60^\circ$. The altitude from the right-angle vertex to the hypotenuse is $3$ units. Find the area of the triangle in square units. Round your answer to two decimal places.
6. 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 $LM$.

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