# Grades 9/10/11 Measurement, Geometry, and Trigonometry

Pythagorean Theorem. Measurement of 2D figures and 3D solids. Optimization. Geometric relationships. Triangle trigonometry. Angles in standard position and trigonometric identities.

This is one of seven strands of the CEMC Grade 9/10/11 courseware. The other strands and more information about this courseware is available on the Grade 9/10/11 homepage.

The alternative formats for some lessons are currently under construction. These will be completed in the coming months.

### Units

## Unit 1: The Pythagorean Theorem, Measurement, and Optimization

In this lesson, we make connections between the geometric and algebraic representations of the Pythagorean Theorem. We will use it to solve applications involving missing side lengths of right triangles and investigate the converse of the Pythagorean Theorem.

In this lesson, we make connections between complex and simple two-dimensional shapes to solve applications involving the perimeter and area of composite shapes.

In this lesson, we will make connections between the net and the surface area of pyramids and cones. We will use different strategies to solve surface area applications.

We continue our exploration of pyramids and cones making connections about their volumes with those of prisms and cylinders. We will solve problems involving the volume of pyramids and cones within different contexts.

In this lesson, we will work with the formulas for the volume and surface area of a sphere to solve application problems.

In this lesson, we will explore how rectangles with the same perimeter may have different areas. We will develop strategies and tools to determine the maximum area of rectangles with a fixed perimeter in different settings.

In this lesson, we will explore how rectangles with the same area may have different perimeters. We deepen our understanding of strategies and tools used for mathematical optimization to determine the minimum perimeter for rectangles with a fixed area.

Conservation of materials and heat are some of the applications of minimizing the surface area of 3D objects. In this lesson, we will make connections between the dimensions and surface area of cylinders and square-based prisms using different strategies to find the optimal shape.

This last optimization lesson makes connections between cylinders and square-based prisms with a fixed surface area and its shape when the volume is maximized.

All of the pencil and paper practice exercises, answers, and solutions for this unit are reproduced here.

This is a collection of additional, and sometimes challenging, problems that extend the material covered in this unit, connect material from different lessons, and further explore real-world applications.

## Unit 2: Geometric Relationships

In this lesson, we will review properties of angles produced by intersecting lines (such as opposite, supplementary, and complementary angles). We will also review properties of angles produced by parallel lines and a transversal.

In this lesson, we will explore the properties and relationships of the interior and exterior angles of triangles, and use these relationships to solve angle problems. The ideas of conjectures and counterexamples are also introduced.

In this lesson, we will explore the properties and relationships of the interior and exterior angles of quadrilaterals and other polygons. We will apply these properties in solving angle problems and to verify conjectures.

In this lesson, we will investigate and describe properties of polygons. These properties will involve either the midpoints of the sides of the polygon, or the diagonals of the polygon.

In this lesson, we will discover some relationships between chords of circles and perpendicular bisectors. We will use the relationships we find to solve problems involving chords and perpendicular bisectors.

In this lesson, we will investigate the area of a sector and arc length. We will also investigate various properties about angles in a circle and use these properties to solve problems.

In this lesson, we will investigate the relationship between a tangent to a circle and the radius drawn to the point of tangency. We hope to discover a relationship and apply the relationship to solve problems.

All of the pencil and paper practice exercises, answers, and solutions for this unit are reproduced here.

This is a collection of additional, and sometimes challenging, problems that extend the material covered in this unit, connect material from different lessons, and further explore real-world applications.

## Unit 3: Trigonometry

In this lesson, we review the concepts of congruence and similarity. We also learn how to compute the scale factor relating similar polygons and use this to solve problems involving perimeter and area.

In this lesson, we review the angle-angle (AA), side-angle-side (SAS), and side-side-side (SSS) similarity rules for demonstrating that two triangles are similar. We next use similar triangle constructions to solve a variety of problems.

In this lesson, we will define the tangent ratio and see how to use it to solve a variety of problems involving right-angled triangles. We will also learn how to use the inverse tangent operation to solve for an interior angle in a right-angled triangle.

In this lesson, we define the sine and cosine ratios and use them to solve a variety of problems involving right-angled triangles. We also learn how to perform calculations involving sine and cosine operations using calculators.

In this lesson, we state and prove the sine law for acute triangles. We then use the sine law to determine side lengths and angles in acute triangles.

In this lesson, we derive the cosine law for acute triangles. We then use the cosine law to determine side lengths and angles in acute triangles.

In this lesson, we review the sine and cosine laws and under what conditions they can be used. We also apply the sine and cosine laws to solve a variety of application-style problems.

In this lesson, we learn how to compute the sine, cosine, and tangent ratios corresponding to oblique angles and vice versa. We see that determining the oblique angle associated with a given sine ratio can give rise to ambiguities. This leads into a discussion of the ambiguous case of the sine law.

In this lesson, we use trigonometric tools developed for two-dimensional right-angled and oblique triangles to solve problems in three-dimensional settings.

All of the pencil and paper practice exercises, answers, and solutions for this unit are reproduced here.

This is a collection of additional, and sometimes challenging, problems that extend the material covered in this unit, connect material from different lessons, and further explore real-world applications.

## Unit 4: Angles in Standard Position and Trigonometric Identities

This lesson introduces angles drawn in standard position in order to extend our understanding of trigonometric ratios beyond acute angles.

In this lesson, we connect angles in standard position to their related acute angles to evaluate trigonometric ratios and find angles. We will also relate negative angles and angles greater than \(360^\circ\) to their coterminal angles between \(0^\circ\) and \(360^\circ\).

This lesson continues our work with trigonometric ratios focusing on special angles where use of a calculator is not needed.

For an acute right triangle, there are six possible ways to make a ratio using two side lengths. In this lesson, we will learn to work with the reciprocal trigonometric ratios and learn about how they behave.

We will identify the fundamental trigonometric identities and learn how to apply these to prove other identities.