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17 problems match “right triangles”. Clear

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.
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.
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.
Right triangle with given circumradius and altitude ●○○○○ 10 ptsA right-angled triangle has circumradius $3$ cm. The altitude drawn to its hypotenuse is $2$ cm. Find the area of the triangle in square centimetres.
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.
Cubed sine at half angle ●○○○○ 10 ptsIf $\theta = 60^\circ$ , find the value of $4\sin^3\frac{\theta}{2}$ . Give your answer as a decimal.
Centre of square on hypotenuse ●●○○○ 20 ptsLet $ABC$ be a right triangle with the right angle at $B$ . Let $ACDE$ be a square drawn outside triangle $ABC$ on side $AC$ . If $M$ is the centre of this square, find $\angle MBC$ in degrees.
Shortest altitude of a triangle ●●○○○ 20 ptsThe sides of a triangle have lengths $15$ , $20$ , and $25$ . Find the length of the shortest altitude.
Area of circle through three points ●●○○○ 20 ptsA circle passes through the points $(1,1)$ , $(1,7)$ , and $(9,1)$ . The area of the circle can be expressed as $k\pi$ . Find the value of $k$ .
Minimum area with given hypotenuse altitude ●●○○○ 20 ptsIn right triangle $ABC$ , the altitude to the hypotenuse has length 4. Find the minimum value of the area of triangle $ABC$ .