options: Analyze, compare, create, and compose shapes.

Prove geometric theorems.

Understand and apply the Pythagorean Theorem.

Define trigonometric ratios and solve problems involving right triangles.

Translate between the geometric description and the equation for a conic section.

Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres).

Apply geometric concepts in modeling situations.

Graph points on the coordinate plane to solve real-world and mathematical problems.

Use coordinates to prove simple geometric theorems algebraically.

Solve real-world and mathematical problems involving area, surface area, and volume.

Understand and apply theorems about circles.

Visualize relationships between two-dimensional and three-dimensional objects.

Understand similarity in terms of similarity transformations.

Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.

Make geometric constructions.

Prove theorems involving similarity.

Understand congruence in terms of rigid motions.

Understand congruence and similarity using physical models, transparencies, or geometry software.

Reason with shapes and their attributes.

Explain volume formulas and use them to solve problems.

Experiment with transformations in the plane.

Draw, construct, and describe geometrical figures and describe the relationships between them.

Apply trigonometry to general triangles.

Draw and identify lines and angles, and classify shapes by properties of their lines and angles.

Find arc lengths and areas of sectors of circles.

Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.

Classify two-dimensional figures into categories based on their properties.
problem_activity: Task
The King of Syracuse reportedly requested Archimedes' advice for determining
if a crown was made with the appropriate mixture of gold and silver.   Archimedes
devised a method requiring the following measurements:
the volume $V$ of water displaced by the crown when it is submerged
in water,
the volume $V_1$ of water displaced by an equal mass (that is, equal to the
mass of the crown)  of gold when
submerged in the water,
the volume $V_2$ of water displaced by an equal mass of silver when
submerged in water.
How much water would a solid gold crown displace?  What about a solid
silver crown?  What if the crown is half gold (by volume) and half silver (by volume)?
If $p$ denotes the percentage of the crown, by volume, which comes from
gold, how much water will the crown displace?
Show that Archimedes' method can always determine the percentage of gold in the crown from its mass and volume.


correct_option_index: 6