Geometry Unit 7: Exploring Circles, Right Triangles, and Trigonometric Ratios

Geometry Unit 7

Six lessons compose geometry unit 7. Students start by reviewing the Pythagorean Theorem and its converse. They then move on to study special right triangles, and trigonometric ratios. Using their newfound knowledge of triangle similarity postulates and theorems, they begin to calculate arc lengths from given sector areas and central angles.

Circles

A circle is a 2-dimensional closed shape that has curved sides. Students learn the definition of a circle, formulas for radius and circumference, and problem-solve related to circles. They also study the properties of chords, secants and tangents within a circle.

Circles are a key topic for developing students’ understanding of relationships between angles and segments. They will analyze inscribed and circumscribed angles, find arc lengths for circles of different sizes, and solve problems relating to central angle, radius, and arc area.

Using the information from the previous lesson, students will investigate the relationship between a radius and a diameter. They will find that a radius is half the diameter and the distance between the centre and the outermost point of the circle, known as the circumference, is double the radius (see figure below).

Right Triangles

A right triangle is a two-dimensional figure with one right angle and three sides. It has a vertex and a hypotenuse, or longest side. The legs are opposite or adjacent to the vertex. The legs can be measured by their lengths, or their angles can be identified and classified as acute, obtuse, or straight.

Students learn about the Pythagorean Theorem and how the lengths of the three sides relate. They also learn that when they know the length of only two sides, they can use sine, cosine, and tangent values to find the remaining side.

Students use tables of known triangle dimensions to compare the sizes of different right triangles. They are challenged to think about unknown angle measures and how they might be solved based on the patterns they have noticed in these tables.

Trigonometric Ratios

In geometry, the ratios of a triangle’s sides to its angles are called trigonometric functions or trig ratios. There are six trigonometric ratios: sine (sin), cosine (cos), tangent (tang), secant, and cosecant (or csc). Students can recall these with the mnemonic SOHCAHTOA.

Students will learn how to choose a trig ratio given an angle and side of a right triangle. They will also see that similar triangles are formed when all three trig ratios are equal.

Today’s Check Your Understanding question asks students to use what they have learned about trig ratios to determine whether two triangles are similar. The question is meant to help students realize that the information they were given in Unit 6 about the ratios of a triangle’s side lengths and angles is encoded in those ratios—they just have to know which trig ratios to select! This is a preview of what they will learn next in lesson 7.4..

Similarity

Students informally understand similarity before starting geometry, but it is formally explored during this unit. Triangle similarity postulates and theorems are analyzed along with side-splitting, proportional medians, angle bisectors, and altitudes.

Two triangles are similar if all corresponding pairs of angles are congruent and the sides that include those angles are proportional to one another. Students also learn how to use transformations to show that a figure is similar to another figure.

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