Lesson 3 | Right Triangles and Trigonometry | 10th Grade Mathematics

Objective

Define the relationship between side lengths of special right triangles.

Common Core Standards

Core Standards

The core standards covered in this lesson

G.SRT.B.4 — Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.

Similarity, Right Triangles, and Trigonometry

G.SRT.B.4 — Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.
G.SRT.B.5 — Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

Similarity, Right Triangles, and Trigonometry

G.SRT.B.5 — Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

Foundational Standards

The foundational standards covered in this lesson

G.CO.C.10

Congruence

G.CO.C.10 — Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

Criteria for Success

The essential concepts students need to demonstrate or understand to achieve the lesson objective

Derive the relationship between side lengths of right triangles with measures of 45°-45°-90° from a square with a diagonal drawn, the properties of base angles of isosceles triangles, and the Pythagorean theorem.
Define the general algebraic relationship between all 45°-45°-90° triangles, citing similarity as the reason for why this relationship can be extrapolated.
Derive the relationship between side lengths of right triangles with angle measures of 30°-60°-90° from an equilateral triangle with an angle bisector drawn from a vertex, and the Pythagorean theorem.
Define the general algebraic relationship between all 30°-60°-90° triangles, citing similarity as the reason for why this relationship can be extrapolated.
Algebraically find the relationship for any side length of special right triangles.

Tips for Teachers

Suggestions for teachers to help them teach this lesson

The following resource is helpful to understand how special right triangles fit in to the common core standards: Mathematical Musings, “Special Right Triangles”
At this point, students are NOT expected to be able to rationalize the denominator or multiply radicals when finding the side lengths of special right triangles. This skill will be developed in the next lesson.

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Anchor Problems

Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

Problem 1

Below is an equilateral triangle with a perpendicular bisector drawn from vertex $$A$$.

Based on this information, what else do you know about the resultant triangles?

Guiding Questions

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Problem 2

Below is a square with diagonal $${\overline {AC}}$$.

Based on this information, what else do you know about the resultant triangles?

Guiding Questions

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Target Task

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

Find the missing side lengths. Show your reasoning.

Additional Practice

The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

Include problems where students are asked to find the actual values of missing legs and hypotenuse of special right triangles.
Include problems where students identify error of relationships between side lengths and hypotenuse of special right triangles.
Include problems that require students to explain their thinking and describe from where the two special right triangles are derived.

EngageNY Mathematics Geometry > Module 2 > Topic D > Lesson 24 — Problem Set and remaining Exercises
Mathematics Vision Project: Secondary Mathematics Three Module 5: Modeling with Geometry — Lesson 5 "Special Rights: A Solidifying Understanding Task" (p. 29 #9-14)

Lesson 2

Lesson 4

Lesson Map

Topic A: Right Triangle Properties and Side-Length Relationships

Define the parts of a right triangle and describe the properties of an altitude of a right triangle.

G.CO.A.1 G.SRT.B.4

Define and prove the Pythagorean theorem. Use the Pythagorean theorem and its converse in the solution of problems.

G.SRT.B.4

Define the relationship between side lengths of special right triangles.

G.SRT.B.4 G.SRT.B.5

Multiply and divide radicals. Rationalize the denominator.

A.SSE.A.2 N.RN.A.2

Add and subtract radicals.

A.SSE.A.2 N.RN.A.2

Topic B: Right Triangle Trigonometry

Define and calculate the sine of angles in right triangles. Use similarity criteria to generalize the definition of sine to all angles of the same measure.

G.SRT.C.6

Define and calculate the cosine of angles in right triangles. Use similarity criteria to generalize the definition of cosine to all angles of the same measure.

G.SRT.C.6

Derive the relationship between sine and cosine of complementary angles in right triangles, and describe sine and cosine as angle measures approach 0°, 30°, 45°, 60°, and 90°.

G.SRT.C.7

Describe and calculate tangent in right triangles. Describe how the value of tangent changes as the angle measure approaches 0°, 45°, and 90°.

G.SRT.C.6 G.SRT.C.7

Solve for missing sides of a right triangle given the length of one side and measure of one angle.

G.SRT.C.8

Topic C: Applications of Right Triangle Trigonometry

Find the angle measure given two sides using inverse trigonometric functions.

G.SRT.C.8

Describe the relationship between slope and the tangent ratio of the angle of elevation/depression. Use the tangent ratio of the angle of elevation or depression to solve real-world problems.

G.SRT.C.8

Solve a modeling problem using trigonometry.

G.SRT.C.8

Topic D: The Unit Circle

Define angles in standard position and use them to build the first quadrant of the unit circle.

F.TF.A.2

Use the first quadrant of the unit circle to define sine, cosine, and tangent values outside the first quadrant.

F.TF.A.2

Topic E: Trigonometric Ratios in Non-Right Triangles

Derive the area formula for any triangle in terms of sine.

G.SRT.D.9

Verify algebraically and find missing measures using the Law of Sines.

G.SRT.D.10

Verify algebraically and find missing measures using the Law of Cosines.

G.SRT.D.10

Use side and angle relationships in right and non-right triangles to solve application problems.

G.SRT.D.11

Right Triangles and Trigonometry

Objective

Common Core Standards

Core Standards

Similarity, Right Triangles, and Trigonometry

Similarity, Right Triangles, and Trigonometry

Foundational Standards

Congruence

Criteria for Success

Tips for Teachers

Anchor Problems

Problem 1

Guiding Questions

Problem 2

Guiding Questions

Target Task

Additional Practice

Lesson Map

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