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17 problems match “right triangles”. Clear

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.
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.
Difference of cone volumes from a triangle ●●○○○ 20 ptsA right-angled triangle with sides 6 cm, 8 cm and 10 cm is revolved about each of the two sides that contain the right angle, giving two solids. Find the difference between their volumes in cm $^3$ . Round to two decimal places.
Smallest modulus on a segment ●●●○○ 50 ptsLet $z$ be a complex number such that $|z - 12| + |z - 5i| = 13$ . Find the smallest possible value of $|z|$ . Round to two decimal places.
Area ratio of faces in a pyramid ●●●○○ 50 ptsThe height of the triangular pyramid $ABCD$ , dropped from vertex $D$ , passes through the point of intersection of the heights of triangle $ABC$ . In addition, it is known that $DB=3$ , $DC=2$ , $\angle BDC=90^{\circ}$ . Find the ratio of the area of face $ADB$ to the area of face $ADC$ .
Area of triangle BDE in space ●●●○○ 50 ptsPoints $A$ , $B$ , $C$ , $D$ , and $E$ are located in 3-dimensional space with $AB= BC= CD= DE= EA= 2$ and $\angle ABC = \angle CDE = \angle DEA = 90^\circ$ . The plane of triangle $ABC$ is parallel to $\overline{DE}$ . What is the area of triangle $BDE$ ?
Hypotenuse from two cone volumes ●●●○○ 50 ptsWhen a right triangle is rotated about one leg, the volume of the cone produced is $800\pi \text{ cm}^3$ . When the triangle is rotated about the other leg, the volume of the cone produced is $1920\pi \text{ cm}^3$ . What is the length (in cm) of the hypotenuse of the triangle?