Determine the coordinates of a point that is a given fractional distance less than one from one end of a line segment to the other in one- and two-dimensional coordinate systems, including finding the midpoint.
Investigate patterns to make conjectures about geometric relationships, including angles formed by parallel lines cut by a transversal, criteria required for triangle congruence, special segments of triangles, diagonals of quadrilaterals, interior and exterior angles of polygons, and special segments and angles of circles choosing from a variety of tools.
Verify theorems about the relationshps in triangles, including proof of the Pythagorean Theorem, the sum of interior angles, basde angles of isosceles triangles, midsegments, and medians, and apply these relationships to solve problems.
Prove a quadrilateral is a parallelogram, rectangle, square, or rhombus using opposite sides, opposite angles, or diagonals and apply these relationships to solve problems.
Apply the definition of similarity in terms of a dilation to identify similar figures and their proportional sides and the congruent corresponding angles.
Apply the Angle-Angle criterion to verify similar triangles and apply the proportionality of the corresponding sides to solve problems.
Prove theorems about similar triangles, including the Triangle Proportionality theorem, and apply these theorems to solve problems.
Identify and apply the relationships that exist when an altitude is drawn to the hypotenuse of a right triangle, including the geometric mean, to solve problems.
Determine the lengths of sides and measures of angles in a right triangle by applying the trigonometric ratios sine, cosine, and tangent to solve problems.
Apply the relationships in special right triangles 30-60-90 and 45-45-90 and the Pythagorean theorem, including Pythagorean triples, to solve problems.
Identify the shapes of two-dimensional cross-sections of prisms, pyramids, cylinders, cones, and spheres and identify three-dimensional objects generated by rotations of two-dimensional shapes.
Determine and describe how changes in the linear dimensions of a shape affect its perimeter, area, surface area, or volume, including proportional and non-proportional dimensional change.
Determine the area of composite two-dimensional figures comprised of a combination of triangles, parallelograms, trapezoids, kites, regular polygons, or sectors of circles to solve problems using appropriate units of measure.
Apply the formulas for the total and lateral surface area of three-diensional figures, including prisms, pyramids, cones, cylinders, spheres, and composite figures to solve problems using appropriate units of measure.
Apply the formulas for the volume of three-dimensional figures, including prisms, pyramids, cones, cylinders, spheres, and composite figures, to solve problems using appropriate units of measure.
Apply theorems about circles, including relationships among angles, radii, chords, tangents, and secants, to solve non-contextual problems.
Apply the proportional relationship between the measure of an arc length of a circle and the circumference of the circle to solve problems.
Apply the proportional relationship between the measure of the area of a sector of a circle and the area of the circle to solve problems.
Show that the eqation of a circle with center at the origin and radius r is x^2 + y^2 = r^2 and determine the equataion for the graph of a circle with radius r and center (h, k), (x - h)^2 + (y - k)^2 = r^2.
Develop strategies to use permutations and combinations to solve contextual problems.
Identify whether two events are independent and compute the probability of the two events occurring together with or without replacement.