This course provides an opportunity for students to explore Trigonometry, Analytic Geometry, and Modeling. Topics include properties and solutions for right triangles, Law of Sines and Cosines and their applications; properties and construction of circles and tangents to circles, properties and construction of cyclic quadrilaterals and regular polygons; parallel and perpendicular lines, deriving their slopes and distance between them; surface area, volume, and perimeter of several types of geometric figures; describing and applying geometric measurements for practical purposes; rotating 2D objects, cross-section of 3D geometric figures; concepts of density and solving design problems using geometric concepts. Students perform constructions, measure figures, observe patterns, discuss their findings, write their own definitions, and formulate and prove geometric conjectures.

**Section 1: Trigonometry of triangle**

**Objectives:**

- To use geometric mean and find lengths of segments in a right triangle
- To determine and use relations in similar right triangles
- Apply the converse of Pythagorean Theorem, to determine is a triangle is right, obtuse or acute
- To define and use six trigonometric ratios as ratios of sides of a right triangle
- To use properties of 45-45-90 degree and 30-60-90 degree triangles to solve problems
- To solve right triangles using trigonometric ratios
- To use trigonometric ratios and solve problems involving angle of elevation and angle of depression
- To find missing angles in a right triangle using inverse trigonometric ratios
- Apply the Law of Sines to solve a triangle
- Solve word problems requiring the Law of Sines
- Be able to determine the number of triangles that exist in the ambiguous case
- Apply the Law of Cosines to solve a triangle
- Solve word problems requiring the Law of Cosines

**Lessons:**

- Properties of Right Triangle
- Trigonometry of Right Triangles
- Special Right Triangles
- Solving Right Triangles
- Law of Sines and its Applications
- Law of Cosines and its Applications

**Section 2: Circles**

**Objectives:**

- To identify segments and lines related to circle
- To identify corresponding central and inscribed angles
- To distinguish types of tangents to circles and apply their properties
- To construct the circumcircle, incircle and excircle of a triangle
- To construct tangents to circle, and common tangent to two circles
- To explore some special classes of quadrilaterals such as tangential and cyclic quadrilaterals
- To determine and construct the locus of set of points that satisfy a given condition
- To construct regular polygons inscribed in a circle

**Lessons:**

- Introduction to Radii, Segments, Lines and Chords as Related to Circles
- Angles and Arcs Related to Circles
- Constructions of an Inscribed and Circumscribed Circles of a Triangle
- Tangent to a Circle
- Construction of Tangents to a Circle
- Cyclic Quadrilaterals
- Construction of Locus
- Constructions of a Regular Polygons Inscribed in a Circle

**Section 3: Analytical geometry in the plane**

**Objectives:**

- To derive an equation of a circle
- To identify the radius and the center of a circle using the equation
- To plot a circle in two-dimensional coordinate system
- To find an equation of line and to express it in different forms
- To determine whether lines parallel, perpendicular, or neither
- To write equations of parallel or perpendicular lines
- To be able to find the angle between two lines
- To determine the coordinates of a point of division of line segment
- To find the perimeter and area of triangle, rectangle, polygon and composite figures

**Lessons:**

- Deriving equation of a circle
- Different Forms of Equations of Lines
- Slope of Parallel Lines
- Slope of Perpendicular Lines
- Angle Between Two Lines
- Distance of a Point from a Given Line and Division of a Line Segment
- Proving Geometric Theorems Algebraically
- Perimeter and Area of Triangle, Rectangle, and Polygon using the Distance Formula

**Section 4: Geometric measurements and dimensions**

**Objectives:**

- To find perimeter and area of polygons
- To find perimeter and area of circle
- To understand units, square units, and cubic units and how they are related to perimeter, area and volume
- To recognize the shapes of prism, cylinder, pyramid, cone and sphere
- Understand the meaning the surface area and volume of three-dimensional figure
- To determine surface area and volume of prism, cylinder, pyramid, cone and sphere
- To state Cavalieri’s Principle and to use it to prove formulas and problems
- To choose appropriate formulas to solve real-life volume and surface area problems

**Lessons:**

- Perimeter and Area of Polygon
- Circumference and Area of a Circle
- Surface Area and Volume of a Prism
- Surface Area and Volume of a Pyramid
- Surface Area and Volume of a Cylinder
- Surface Area and Volume of a Cone
- Surface Area and Volume of a Sphere

** Section 5: Modeling with geometry**

** Objectives:**

- Model geometric, two-dimensional and three-dimensional shapes by real life examples;
- To determine volume of prism, cylinder, pyramid, cone and sphere after doubling, tripling, or halving a dimension;
- Identify the shapes of two-dimensional cross sections of three-dimensional objects;
- Identify three-dimensional objects generated by rotations of two-dimensional objects;
- To determine density of a three-dimensional object;
- To determine the density of population;
- To solve design problems using geometrical methods and knowledge.

**Lessons:**

- Application of the Volumes of Cones, Cylinders, Pyramids and Spheres
- Rotating 2D Objects
- Cross-Sections of Three-Dimensional Objects
- Describing an Object Using Geometrical Concepts
- Applying Concepts of Density in Modeling Situations
- Solving Design Problems Using Geometrical Methods