options: Use coordinates to prove simple geometric theorems algebraically.

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

Prove geometric theorems.

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

Experiment with transformations in the plane.

Understand and apply theorems about circles.

Classify two-dimensional figures into categories based on their properties.

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

Prove theorems involving similarity.

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

Apply trigonometry to general triangles.

Understand similarity in terms of similarity transformations.

Understand congruence in terms of rigid motions.

Make geometric constructions.

Apply geometric concepts in modeling situations.

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

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

Define trigonometric ratios and solve problems involving right triangles.

Explain volume formulas and use them to solve problems.

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

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

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

Understand and apply the Pythagorean Theorem.

Find arc lengths and areas of sectors of circles.

Analyze, compare, create, and compose shapes.

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

Reason with shapes and their attributes.
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: 14