Curriculum / Math / 10th Grade / Unit 4: Right Triangles and Trigonometry / Lesson 2
Math
Unit 4
10th Grade
Lesson 2 of 19
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Define and prove the Pythagorean theorem. Use the Pythagorean theorem and its converse in the solution of problems.
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.
The foundational standards covered in this lesson
8.EE.A.2 — Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.
8.G.B.6 — Explain a proof of the Pythagorean Theorem and its converse.
8.G.B.7 — Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
8.G.B.8 — Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
The essential concepts students need to demonstrate or understand to achieve the lesson objective
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Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding
25-30 minutes
Given the subdivided right triangle below, show that $${a^2+b^2=c^2}$$.
Geometry > Module 2 > Topic D > Lesson 24 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..
Find $${RS}$$.
8.3 Using Similar Right Triangles is made available by CK-12 under the CC BY-NC 3.0 license. © CK-12 Foundation 2017. Visit them at ck12.org. Accessed Feb. 28, 2018, 11:10 a.m..
A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved
5-10 minutes
In each row in the table below are the lengths of the legs and hypotenuse of different right triangles. Find the missing side lengths in each row in simplest radical form.
In right triangle $${ABC}$$ with $${\angle C}$$ a right angle, an altitude of length $$h$$ is dropped to side $${\overline{AB}}$$ into segments of length $$x$$ and $$y$$. Use the Pythagorean theorem to show $$h^2=xy$$.
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.
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Define the relationship between side lengths of special right triangles.
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.
Standards
G.CO.A.1G.SRT.B.4
G.SRT.B.4
G.SRT.B.4G.SRT.B.5
Multiply and divide radicals. Rationalize the denominator.
A.SSE.A.2N.RN.A.2
Add and subtract radicals.
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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.
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.6G.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.
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.
Solve a modeling problem using trigonometry.
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.
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.
Use side and angle relationships in right and non-right triangles to solve application problems.
G.SRT.D.11
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