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5 problems match “equilateral triangle”. Clear

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1. Altitude of an equilateral triangle ●○○○○ 10 ptsAn equilateral triangle has side length 6 cm. Find the length of its altitude. Round to two decimal places.
2. Height of a pile of three pipes ●●○○○ 20 ptsThree identical cylindrical pipes, each with diameter $10\text{ cm}$, are stacked so that each pipe touches the other two: two lie side by side on the ground and the third rests on top of them, touching both. Let $h$ be the height of this pile, from the ground to the top of the highest pipe. Find $h$ in cm. Round to two decimal places.
3. Angle in pentagon with inner triangle ●●○○○ 20 ptsIn a regular pentagon $ABCDE$, an equilateral triangle $ABM$ is drawn so that $M$ lies inside the pentagon. Determine the size of the angle $BCM$ in degrees.
4. Sums of squares on a unit circle ●●●○○ 50 ptsThree points $A(x_1, y_1)$, $B(x_2, y_2)$, $C(x_3, y_3)$ lie on the unit circle $x^2+y^2=1$ and satisfy $x_1+x_2+x_3=0$ and $y_1+y_2+y_3=0$. Find $x_1^2+x_2^2+x_3^2$ and $y_1^2+y_2^2+y_3^2$.
5. Centroid distance to a plane ●●●○○ 50 ptsThe vertices $A$, $B$, $C$ of an equilateral triangle $ABC$ with side length $6$ are at distances $1$, $2$ and $3$ from a plane $\alpha$. Find all possible distances from the centroid $G$ of triangle $ABC$ to the plane $\alpha$. Round each value to two decimal places and give them in increasing order.