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Select the cluster for resources on this page:

**Standards**:

- SRT-A.1. Verify experimentally the properties of dilations given by a center and a scale factor:
- SRT-A.1.a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.
- SRT-A.1.b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
- SRT-A.2. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
- SRT-A.3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

**Technology-enhanced investigations:**

Mathwords.com: Math Dictionary: Key vocabulary for this domain. Use with SRT-A, SRT-B, SRT-C, and SRT-D:

LearnZillion:

- Lesson set: Verify experimentally the properties of dilations given by a center and a scale factor: Five video lessons: Understand that a dilation takes a line not passing through the center to a parallel line, draw an image of a dilation by using the center of dilation and scale factor, calculate the scale factor of a dilation, draw an image of dilation with a negative scale factor, solve word problems by doing dilations. Aligns with SRT-A.1.
- Lesson set: Decide if two figures are similar and explain the meaning of triangle similarity: Four video lessons: Determine if two figures are similar by using transformations and dilations, understand that a dilation of a triangle creates a similar triangle, determine triangle similarity using properties of similarity transformations, find missing measures in similar figures. Aligns with SRT-A.2.
- Lesson set: Use the properties of similarity transformations to establish the AA criterion: Five video lessons: Show triangle similarity using translation, by using translation and dilation, by checking for corresponding congruent angles; prove AA similarity by overlapping angles; understand why the third angles of similar triangles are congruent. Aligns with SRT-A.3.

National Library of Virtual Manipulatives: Dilations (Java required)

Southeast Comprehensive Center of SEDL: Video: Understand similarity in terms of similarity transformations: This video aligns with standard SRT-A.2. View online or download.

Wolfram Demonstrations Project: Download the free Wolfram CDF player to interact with the following manipulatives. Note: Within the Wolfram Demonstration Project are 5 manipulatives addressing SRT-A.1, 16 manipulatives for STR-A.2, and 2 manipulatives for SRT-A.3. Among those:

- Enlargement in two dimensions: Dilations. Aligns with SRT-A.1.
- Dividing a Right Triangle by the Altitude to the Hypotenuse: The two triangles formed are similar to the original triangle and to each other. Aligns with SRT-A.2. SRT-A.3.
- Einstein's most excellent proof: "a new proof of the Pythagorean theorem based on the properties of similar triangles." Aligns with SRT-A.2, SRT-B.4.

**Multiple Choice:**

CK-12:
Dilations in the Coordinate Plane: Practice questions with review
content and video. Review content includes a *Types of
Transformations Study Guide*, which is an overview of reflection,
rotation, dilation, and translation. Aligns with CO-A.2 and SRT-A.1.

Khan Academy: Practice questions with videos.

- Similarity: Unit with quizzes. Aligns with SRT-A.2, SRT-A.3.
- Determine similar triangles 1: Using Angles, AA. Aligns with SRT-A.2, SRT-A.3.
- Determine similar triangles 2: Using Sides, SSS. Aligns with SRT-A.2.

MathBitsNotebook: Geometry: Similarity includes lessons, then practice problems on dilations, dilations and lines, similarity (numerical problems), similarity measurements, side splitter theorem, angle bisector theorem, and similarity proofs. Use with SRT-A.1, SRT-A.2, SRT-A.3, SRT-B.4, and SRT-B.5.

MathsIsFun.com: Resizing: Dilation or resizing is defined and illustrated. Learners then use a virtual manipulative to experiment with dilations involving a triangle, rectangle, and L-shape, selecting a scale factor and a center. Following this experiment are multiple choice questions on resizing. Aligns with SRT-A.1.

**Constructed-response:**

AlgebraLab.org: Similar Triangles: Lesson and fill in the blank practice problems, including word problems. Be sure to show the related AlgebraLab documents for additional practice problems.

Khan Academy: Practice questions with videos.

- Dilations: Learners construct; and enter their answers based on the construction. Aligns with SRT-A.1.

**Performance tasks:**

Illustrative Mathematics: Geometry:

- Dilating a line: Aligns with SRT-A.1.
- Are they similar?: Aligns with SRT-A.2.
- Similar triangles: Aligns with SRT-A.3.

Inside Mathematics: MARS Tasks:

- Rhombuses: Students use properties of the rhombus and Pythagorean theorem, and prove similarity of rhombuses. Aligns with standards SRT-A.2, SRT-B.5.
- Hopewell Geometry: Students work with the Pythagorean theorem, find angles using trig ratios, and prove similarity involving triangles. Aligns with standards SRT-A.2, SRT-B.5, and SRT-C.6, SRT-C.7, SRT-C.8.

Mathematics Assessment Project: High School: Geometry:

- Task 222: Geometry Problems: Circles and Triangles
- Task 452: Analyzing Congruence Proofs

Mathematics Vision Project, Secondary 2 Student Edition:

- Module 6: Similarity & Right Triangle Trigonometry: This module contains 11 classroom tasks. Task 1 addresses standard SRT-A.1, task 2 SRT-A.2 and SRT-B.4; task 3 SRT-A.2 and SRT-A.3.

**Standards**:

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

**Technology-enhanced investigations:**

Cut the Knot: Pythagorean Theorem and its Many Proofs: Written proofs.

GeoGebra Book of Proofs Without Words for the Pythagorean Theorem by Steve Phelps is a series of manipulatives for exploration.

National Library of Virtual Manipulatives: investigate methods for determining congruent triangles (Java required). Explore SSS, SAS, ASA, SSA. Note SSA does not guarantee congruent triangles.

Proofs of Pythagoras' Theorem Using Similarity: Aligns with SRT-B.4.

Wolfram Demonstrations Project: Download the free Wolfram CDF player to interact with the following manipulatives. Note: Within the Wolfram Demonstration Project are 10 manipulatives addressing SRT-B.4, 1 manipulative for STR-B.5. Among those:

- Angle Bisector Theorem: Aligns with SRT-B.4.
- Euler line: This line passes through the incenter, circumcenter, and orthocenter of a triangle. Aligns with SRT-B.4.
- The ratio theorem: Uses a parallel line to one side of a triangle that intersects the other two sides. Aligns with SRT-B.4, SRT-B.5.
- How tall is that building? Uses similar triangles to find the height of a building. Aligns with SRT-B.4, SRT-C.6.
- Einstein's most excellent proof: "a new proof of the Pythagorean theorem based on the properties of similar triangles." Aligns with SRT-A.2, SRT-B.4.

**Multiple Choice:**

Khan Academy: Practice questions with videos.

- Solving problems with similar and congruent triangles: Aligns with SRT-B.5.

MathBitsNotebook: Geometry:

- Similarity includes lessons, then practice problems on dilations, dilations and lines, similarity (numerical problems), similarity measurements, side splitter theorem, angle bisector theorem, and similarity proofs. Use with SRT-A.1, SRT-A.2, SRT-A.3, SRT-B.4, and SRT-B.5.
- Right Triangles includes lessons, then practice problems on the geometric mean in right triangles, Pythagorean theorem, and special right triangles. Use with SRT-B.4, SRT-B.5, SRT-C.6, and SRT-C.8.

**Constructed-response:**

Khan Academy: Practice questions with videos.

- Solving similar triangles 1 (Basic): Given side lengths, find missing side length. Aligns with SRT-B.5.
- Solving similar triangles 2 (Advanced): Given side lengths, find missing side length. Aligns with SRT-B.5.

**Performance tasks:**

Illustrative Mathematics: Geometry:

- Joining two midpoints of the sides of a triangle: Aligns with SRT-B.4.
- Bank Shot: Aligns with SRT-B.5.
- Extensions, bisections, and dissections in a rectangle: Aligns with SRT-B.5.
- Congruence of parallelograms: Aligns with CO-C.11 and SRT-B.5.

Inside Mathematics: MARS Tasks:

- Rhombuses: Students use properties of the rhombus and Pythagorean theorem, and prove similarity of rhombuses. Aligns with standards SRT-A.2, SRT-B.5.
- Hopewell Geometry: Students work with the Pythagorean theorem, find angles using trig ratios, and prove similarity involving triangles. Aligns with standards SRT-A.2, SRT-B.5, and SRT-C.6, SRT-C.7, SRT-C.8.

Mathematics Assessment Project: High School: Geometry:

- Task 419: Proofs of the Pythagorean Theorem
- Task 428: Modeling: Rolling Cups
- Task 429: Solving Geometry Problems: Floodlights

Mathematics Vision Project, Secondary 2 Student Edition:

- Module 6: Similarity & Right Triangle Trigonometry: This module contains 11 classroom tasks. Task 2 addresses standards SRT-A.2 and SRT-B.4; tasks 4-5 and task 7 include standards SRT-B.4, SRT-B.5.

NCTM's Reasoning and Sense Making Task Library: Bank Shot includes the task overview, teacher notes for its use in the classroom, and student activity sheet. Aligns with CO-B.6, CO-C.9, SRT-B.5, and mathematical practice standards 1, 2, 3, and 7.

**Standards**:

- SRT-C.6. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
- SRT-C.7. Explain and use the relationship between the sine and cosine of complementary angles.
- SRT-C.8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

**Technology-enhanced investigations:**

**
Are you wondering why a**^{2} **+ b**^{2}** = c**^{2}** in the right triangle?**

See an easy proof from the MacTutor History of Mathematics archive. Also view the GeoGebra Book of Proofs Without Words for the Pythagorean Theorem by Steve Phelps, which is a series of manipulatives.

GCSE Maths Tutor: Introducing Trigonometry: Videos 9-13 on Pythagorean theorem, sine, cosine, and tangent ratios for the right triangle, which align with SRT-C:

- #9 - Pythagoras'Theorem - Part 1: "This video looks at the Pythagoras' Theorem. An appreciation of the theorem is given graphically and algebraically. Following is a formal proof, explained in some detail. Before problems are attempted, a little time is spent on 'rounding to decimal places'. The video ends with three worked examples on the Pythagoras Theorem."
- #10 - Pythagoras'Theorem - Part 2: "This video is the
second of two about Pythagoras' Theorem. Another proof is described.
This one uses similar triangles.

The concept of 'triples' (right-angled triangles with integral number sides) is explored. - #11 - The Sine Ratio: "The Sine Ratio is defined and its limits explored. The sine values of common angles 30, 45 and 60 degrees are described The tutorial includes worked examples of three common problem types."
- #12 - The Cosine Ratio: "The cosine is defined and its limits explored. The values of common angles 30, 45 and 60 degrees are described. The tutorial includes worked examples of three common problem types."
- #13 - The Tangent Ratio: "The tangent ratio is defined and its limits explored. The values of common angles 30, 45 and 60 degrees are described. The tutorial includes worked examples of three common problem types."

LearnZillion:

- Lesson set: Understand side ratios in right triangles are properties of the angles; define trigonometric ratios for acute angles of a right triangle: Six video lessons: Find the sine, cosine, tangent, cosecant, secant, and cotangent for acute angles in a right triangle. Aligns with SRT-C.6.
- Lesson set: Relate side ratios to angles and derive trigonometric ratios: Five video lessons: Explore similarity through dilation and translation, understand that similar triangles share angle measures and side ratios; find sine, cosine, tangent values using side ratios. Aligns with SRT-C.6.
- Lesson set: Relate the sine and cosine of complementary angles: Five video lessons: Relate sine and cosine by analyzing graphs, by examining triangles; simplify basic trig expressions using sine and cosine. Aligns with SRT-C.7.
- Lesson set: Understand the relationship between the sine and cosine of complementary angles: Six video lessons: Acute angles of a right triangle are complementary, graph circles using Pythagorean theorem; find coordinates for 30, 60, 45 degree angles of a unit circle; find all trig coordinates on a unit circle, relate sine and cosine values on a unit circle. Aligns with SRT-C.7.
- Lesson set: Solve problems using trigonometric ratios and the Pythagorean Theorem: Six video lessons: Solve problems, model using angles of elevation and depression, use inverse trig functions, find angles of elevation and depression; solve word problems, model word problems using trig ratios and Pythagorean Theorem. Aligns with SRT-C.8.
- Lesson set: Solve right triangles using trigonometric ratios and the Pythagorean Theorem: Six video lessons: Find right triangle angle measures using inverse sine, inverse cosine, and inverse tangent; solve non-right triangles using Law of Sines, Law of Cosines; apply trigonometry and Pythagorean Theorem in context. Aligns with SRT-C.8.

MIT BLOSSOMS: Video lesson with additional teacher and learner resources. Description is from the video summary. The Pythagorean Theorem: Geometry's Most Elegant Theorem: "This lesson teaches students about the history of the Pythagorean theorem, along with proofs and applications." Aligns with CO-A.1 and SRT-C.6, SRT-C.8.

National Library of Virtual Manipulatives: (Java required)

- Walk through a visual proof of the Pythagorean theorem.
- Right Triangle Solver: Practice using Pythagorean Theorem and definitions of the trig functions to find sides and angles in right triangles.

Ohio Resource Center on YouTube: Tutorials for High School Mathematics: Video 22: Right Triangle Trig. A review of trig definitions and applying trig in practical problems.

Wolfram Demonstrations Project: Download the free Wolfram CDF player to interact with the following manipulatives. Note: Within the Wolfram Demonstration Project are 3 manipulatives addressing SRT-C.6, 1 manipulative for STR-C.7, and 7 manipulatives for STR-C.8. Among those:

- How tall is that building? Uses similar triangles to find the height of a building. Aligns with SRT-B.4, SRT-C.6.
- Sine, Cosine, and Tangent Using Ratios of Sides of a Right-Angled Triangle: Aligns with SRT-C.6.
- SOHCAHTOA Triangle: Aligns with SRT-C.6.
- Cofunction identities for sine and cosine: Aligns with SRT-C.7.
- Trig functions for a right triangle: Aligns with SRT-C.8, TF-C.8.
- Ball in V Groove geometry: Aligns with SRT-C.8.
- Moving a couch around a corner: Aligns with SRT-C.8, MG-A.3.
- Two wheel belt: Trig problem to find the length of a belt wrapped around two wheels. Aligns with SRT-C.8.
- Person walking away from a spotlight: Aligns with SRT-C.8, SRT-D.11.

**Multiple Choice:**

CK-12: Geometry: Video examples, practice problems, and real world applications. The following align with SRT-C.8:

- Distance and Triangle Classification using the Pythagorean Theorem: Find missing sides to calculate the area.
- Trigonometric Ratios with a Calculator: Trigonometric ratios based on sides of right triangles in relation to an angle.
- Solve right triangles using inverse trigonometric functions
- Trigonometry Word Problems

MathBitsNotebook: Geometry:

- Right Triangles includes lessons, then practice problems on the geometric mean in right triangles, Pythagorean theorem, and special right triangles. Use with SRT-B.4, SRT-B.5, SRT-C.6, and SRT-C.8.
- Trigonometry: Right Triangles includes lessons, then practice problems on trigonometry-numerical (sides and angles), sine and cosine of complementary angles, word problems, and problems "needing" trigonometry. Use with SRT-C.6, SRT-C.7, and SRT-C.8.

**Constructed-response:**

AlgebraLab.org: Lesson and fill in the blank practice problems.

- Pythagorean Theorem lesson and exercises
- Word problems with Pythagorean Theorem

Khan Academy: Right Triangles and Trigonometry unit. Practice questions with videos.

- Trig ratios in right triangles: Given side lengths, find sine, cosine, or tangent of an acute angle. Aligns with SRT-C.6, SRT-C.7.
- Solve for a side in right triangles: Given a side and and acute angle, find another side using trig. Aligns with SRT-C.6, SRT-C.7.
- Solve for an angle in right triangles: Given two sides, find one of the acute angles using trig. Aligns with SRT-C.6, SRT-C.7.
- Right triangle trigonometry word problems: Applying right triangles. Aligns with SRT-C.7, SRT-C.8.

Mathguide.com: Pythagorean theorem quizmaster. The following are fill in and check answers, which are rounded to the nearest tenth. The formula is presented with the following:

- Find the hypotenuse
- Find a leg in a right triangle.

**Performance tasks:**

Illustrative Mathematics: Geometry:

- Defining Trigonometric Ratios: Aligns with SRT-C.6.
- Sine and Cosine of Complementary Angles: Aligns with SRT-C.7.

Inside Mathematics: MARS Task: Hopewell Geometry: Students work with the Pythagorean theorem, find angles using trig ratios, and prove similarity involving triangles. This task aligns with standards SRT-A.2, SRT-B.5, and SRT-C.6, SRTC.7, SRT-C.8.

Mathematics Assessment Project: High School: Geometry:

- Task 216: Calculating Volumes of Compound Objects
- Task 403: Inscribing and Circumscribing Right Triangles
- Task 222: Geometry Problems: Circles and Triangles
- Task 432: Solving Quadratic Equations: Cutting Corners

Mathematics Vision Project, Secondary 2 Student Edition:

- Module 6: Similarity & Right Triangle Trigonometry: This module contains 11 classroom tasks. Standards SRT-C.6, SRT-C.7, and SRT-C.8 are found within tasks 8-11.

**Standards**:

- SRT-D.9. (+) Derive the formula
*A = 1/2 ab sin(C)*for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. - SRT-D.10. (+) Prove the Laws of Sines and Cosines and use them to solve problems.
- SRT-D.11. (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces)

**Technology-enhanced investigations:**

CalculatorSoup:

- Law of Sines Calculator. Aligns with SRT-D.11.
- Law of Cosines Calculator. Aligns with SRT-D.11.

Dave's Short Trig Course:

- Area of a
Triangle: Derivation of
*A = 1/2 ab sin(C).* - Laws of Cosines and Sines: Explanation and proofs.

GCSE Maths Tutor: Introducing Trigonometry: Videos 14-16 align with SRT-D:

- #14 The Sine Rule: The Law of Sines is described and the tutorial includes worked examples of two common problem types.
- #15 The Cosine Rule: The Law of Cosines is described and the tutorial includes worked examples of two common problem types.
- #16-
The Area of a Triangle: "Two different methods for calculating
the area of a triangle are explained.

The video includes worked examples of two common problem types."

Khan Academy:

- Proof of Law of Sines: video. Aligns with SRT-D.10.
- Proof of Law of Cosines: video. Aligns with SRT-D.10.

National Library of Virtual Manipulatives: Triangle Solver: Practice using Law of Sines and Law of Cosines.

Wolfram Demonstrations Project: Download the free Wolfram CDF player to interact with the following manipulatives. Note: Within the Wolfram Demonstration Project are 4 manipulatives addressing SRT-D.10, and 5 manipulatives for STR-D.11. Among those:

- Law of Sines: Aligns with SRT-D.10, SRT-D.11.
- Law of Cosines: Aligns with SRT-D.10, SRT-D.11.
- Hunger's Law of Cosines dissection: Visual proof. Aligns with SRT-D.10.
- Calculating the distance between two inaccessible points: Aligns with SRT-D.10, SRT-D.11.
- Solving oblique triangles: Aligns with SRT-D.11
- Person walking away from a spotlight: Aligns with SRT-C.8, SRT-D.11.

**Multiple Choice:**

MathBitsNotebook: Geometry: Trigonometry: Oblique Triangles includes lessons, then practice problems on areas of triangles using trigonometry, law of sines, law of cosines, navigation and vectors. Use with SRT-D.9, SRT-D.10, and SRT-D.11.

MathsIsFun.com: Tutorials on the Law of Sines and Law of Cosines. Examples are shown. Learners then complete 10 multiple choice problems on each law. Aligns with SRT-D.10 and SRT-D.11.

**Constructed-response:**

Khan Academy: Non-right Triangles and Trigonometry unit. Practice questions with videos.

- Solve triangles using Law of Cosines: Aligns with SRT-D.10, SRT-D.11.
- Solve triangles using Law of Sines: Aligns with SRT-D.10, SRT-D.11.
- General triangle word problems: Law of Sines and Law of Cosines word problems: Aligns with SRT-D.10, SRT-D.11.

**Performance tasks:**

Georgia Standards of Excellence Framework: Pre-Calculus Unit 3: Trigonometry of General Triangles. This unit contains five tasks: Finding a New Area Formula for Triangles, Proving the Law of Cosines, Proving the Law of Sines, The Hinge Theorem, and a Culminating Task: Combining Lots. Tasks align to SRT-D.9, SRT-D.10, SRT-D.11, and are also related to SRT-C.6, SRT-C.7, and SRT-C.8.

Mathematics Vision Project, Secondary 3 Student Edition:

- Module 5: Modeling with Geometry: This module contains 8 classroom tasks. Multiple standards are addressed. Standards SRT-D.10 and SRT-D.11 are found in tasks 4-7.

Common Core Math:
Intro | HS Geometry Domain: CO | **SRT** | C | GPE | GMD | MG |