options: Define trigonometric ratios and solve problems involving right triangles.

Understand and apply the Pythagorean Theorem.

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

Understand congruence in terms of rigid motions.

Analyze, compare, create, and compose shapes.

Apply geometric concepts in modeling situations.

Make geometric constructions.

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

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

Find arc lengths and areas of sectors of circles.

Experiment with transformations in the plane.

Prove geometric theorems.

Explain volume formulas and use them to solve problems.

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

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

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

Understand and apply theorems about circles.

Apply trigonometry to general triangles.

Understand similarity in terms of similarity transformations.

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

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

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

Prove theorems involving similarity.

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

Use coordinates to prove simple geometric theorems algebraically.

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

Reason with shapes and their attributes.
problem_activity: Task
Global Positioning System or GPS devices receive input from satellites and use
this information to locate our position on the planet.   The information received from
each individual satellite gives the distance from the GPS  device to that satellite and the location of the satellite.  The set of points at a fixed distance from a satellite form
a sphere so when the GPS receives its distance from a given satellite, this tells
us that it lies on a particular sphere.  Data from several satellites will
locate the GPS device on the intersection of spheres.  This problem examines different scenarios for intersections of spheres from the point of view of the GPS device: how many
different satellites are needed to locate the GPS device?  Does it matter how
the satellites are configured in space?
Below is a sample picture showing three satellites and their distances, $d_1, d_2, d_3$ from a point on the earth:
[IMG]
The three parts of this problem investigate different positions of the three
satellites pictured above: the goal is to find scenarios where one, two or three of the distances
$d_1$, $d_2$, $d_3$ are (or are not) sufficient to locate the GPS unit.
Give an example of a satellite and GPS location so that the GPS device can determine its location on the earth with input
from this single satellite.  Also give an example when this one signal does not provide
enough information  to designate a location on the earth.
Give  an example of a configuration of two satellites  and the GPS so that the GPS device can determine its location on the earth with input
from the two satellites (but not from either alone).   Also give an example where these two signals do not provide
enough information  to designate a location on the earth.
Give  an example of a configuration of three satellites and the GPS so that  the GPS device can determine its location on the earth with input
from three satellites (but not from any pair of the satellites).   Also give an example where three signals do not provide
enough information  to designate a location on the earth.


correct_option_index: 5

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