91Ó°ÊÓ

Use the following information. The mean of a set of data is an average value of the data. Suppose \(\triangle A B C\) has vertices \(A(16,8), B(2,4),\) and \(C(-6,12)\) Make a conjecture about the centroid and the means of the coordinates of the vertices.

A right prism has square bases with edges that are three times as long as the lateral edges. The prism's total area is \(750 \mathrm{m}^{2}\). Find the volume.

State whether each sentence is always, sometimes, or never true. Justify your reasoning. The three medians of a triangle intersect at a point inside the triangle.

A sphere is inscribed in a cone with radius \(6 \mathrm{cm}\) and height \(8 \mathrm{cm} .\) Find the volume of the sphere.

A cone is inscribed in a regular square pyramid with slant height \(9 \mathrm{cm}\) and base edge \(6 \mathrm{cm} .\) Make a sketch. Then find the volume of the cone.

A right prism has height \(x\) and bases that are equilateral triangles with sides \(x .\) Show that the volume is \(\frac{1}{4} x^{3} \sqrt{3}\)

A right prism has height \(h\) and bases that are regular hexagons with sides \(s .\) Show that the volume is \(\frac{3}{2} s^{2} h \sqrt{3}\)

REASONING Compare and contrast a perpendicular bisector and a median of a triangle.

REASONING Find a counterexample to the statement An altitude and an angle bisector of a triangle are never the same segment.

A \(120^{\circ}\) sector is cut out of a circular piece of tin with radius 6 in. and bent to form the lateral surface of a cone. What is the volume of the cone?