Geometry



Solving Problems with Right Triangles

The Lesson Activities will help you meet these educational goals:

• Content Knowledge—You will use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

• Mathematical Practices—You will make sense of problems and solve them, use mathematics to model real-world situations, and use appropriate tools strategically.

• Inquiry—You will perform an investigation in which you will make observations, analyze results and communicate your results in tables and written form, and draw conclusions.

• 21st Century Skills—You will employ online tools for research and analysis, assess and validate information, and communicate effectively.

Directions

You will evaluate some of these activities yourself, and your teacher may evaluate others. Please save this document before beginning the lesson and keep the document open for reference during the lesson. Type your answers directly in this document for all activities.

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Self-Checked Activities

Read the instructions for the following activities and type in your responses. At the end of the lesson, click the link to open the Student Answer Sheet. Use the answers or sample responses to evaluate your own work.

1. Right Triangles and the Pythagorean Theorem

Given any kind of triangle, you can find its side lengths by applying the Pythagorean Theorem for right triangles, depending on what information you’re given.

You will use GeoGebra to see how the Pythagorean Theorem can be used to solve non-right triangles. Go to right triangles and the Pythagorean Theorem , and complete each step below. If you need help, follow these instructions for using GeoGebra.

a. Create a line through point B perpendicular to [pic]and label the intersection of the perpendicular line and [pic]point D. Measure and record BD and m[pic]. What do you know about ∆ABD and ∆BCD based on their angle measurements? (If you accidentally move point B before taking your measurements, use the “Move B back to initial position” button.)

Type your response here:

b. Use the Pythagorean Theorem to find AD and DC. Then find AC using addition. Show your calculations.

Type your response here:

c. Verify your calculations in part b by displaying the lengths AD and AC in GeoGebra. Then select point B and move it around to different locations. As point B moves, the side lengths of the triangles change; however, notice that ∆ABC continues to comprise two right triangles.

Record the side lengths in the table for four different positions of point B. (Complete both parts of the table for each position of B: the value of BD will be the same in the corresponding rows of both parts of the table.) Use the Pythagorean Theorem to verify the values of AD and DC by filling in and comparing the last two columns in the table.

Type your response here:

|AB |AD |BD |AB2 |AD2 + BD2 |

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|BC |DC |BD |BC2 |DC2 + BD2 |

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How did you do? Check a box below.

Nailed It!—I included all of the same ideas as the model response on the Student Answer Sheet.

Halfway There—I included most of the ideas in the model response on the Student Answer Sheet.

Not Great—I did not include any of the ideas in the model response on the Student Answer Sheet.

2. Inverse Trigonometric Functions

You will use GeoGebra to practice using trigonometric functions and their inverses. Go to inverse trig functions, and complete each step below.

a. Click the refresh button until you get a right triangle with [pic]as the right angle. Fill in the fraction forms of sin B, cos B, tan B and then the associated measure of [pic] using sin-1, cos-1, and tan-1, rounded to three decimal places each. Then fill in the fraction forms of sin C, cos C, tan C, and the associated measure of [pic] using sin-1, cos-1, and tan-1 rounded to three decimal places.

Type your response here:

|p = sin B [pic] |q = cos B |r = tan B [pic] |m[pic] = |m[pic] = |m[pic] = |

| |[pic] | |sin-1 p |cos-1 q |tan-1 r |

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|x = sin C |y = cos C |z = tan C |m[pic] = |m[pic] = |m[pic] = |

|[pic] |[pic] |[pic] |sin-1 x |cos-1 y |tan-1 z |

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b. What do you observe about the angle measurements despite the different inverse functions used? Show that the sum of the interior angles of the triangle is 180°.

Type your response here:

c. Click the refresh button until you get a right triangle with [pic]as the right angle, and fill in the table below as you did in part a.

Type your response here:

|p = sin A |q = cos A |r = tan A [pic] |m[pic] = |m[pic] = |m[pic] = |

|[pic] |[pic] | |sin-1 p |cos-1 q |tan-1 r |

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|x = sin C |y = cos C |z = tan C |m[pic] = |m[pic] = |m[pic] = |

|[pic] |[pic] |[pic] |sin-1 x |cos-1 y |tan-1 z |

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d. What do you observe about the angle measurements despite the different inverse function used? Show that the sum of the interior angles of the triangle is 180 degrees.

Type your response here:

e. Click the refresh button until you get a right triangle with [pic]as the right angle, and fill in the table below as you did in parts a and c.

Type your response here:

|p = sin A |q = cos A |r = tan A |m[pic] = |m[pic] = |m[pic] = |

|[pic] |[pic] |[pic] |sin-1 p |cos-1 q |tan-1 r |

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|x = sin B |y = cos B |z = tan B |m[pic] = |m[pic] = |m[pic] = |

|[pic] |[pic] |[pic] |sin-1 x |cos-1 y |tan-1 z |

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f. What do you observe about the angle measurements despite the inverse function used? Show that the sum of the interior angles of the triangle is 180°.

Type your response here:

How did you do? Check a box below.

Nailed It!—I included all of the same ideas as the model response on the Student Answer Sheet.

Halfway There—I included most of the ideas in the model response on the Student Answer Sheet.

Not Great—I did not include any of the ideas in the model response on the Student Answer Sheet.

3. Right Triangles and Trigonometric Ratios

You will use GeoGebra to find the measurements of William’s kite using right triangles and trigonometric ratios. Go to right triangles trigonometric ratios, and complete each step below.

a. In GeoGebra, you should see a sketch of William’s kite. You’ll use the sketch to find the lengths of the dowels needed to make the kite. William has indicated that he would like the kite to be AD = 120 cm wide across the bottom, which is labeled on the sketch. Using the width of the kite and the fact that the kite is made from two congruent right triangles, what is AC?

Type your response here:

b. The instructions in the kite book state that sin(ABC = [pic]. Since the two right triangles that make up the kite are congruent, then sin(CBD = [pic] as well. Using the definition of sine, what is AB? Show your work.

Type your response here:

c. Using inverse trigonometric functions, what is m[pic]?

Type your response here:

d. Using m[pic], find m[pic]. Show your work.

Type your response here:

e. Let x represent BC. Using either [pic] or [pic] and trigonometric functions, solve for x to three decimal places in at least three different ways. Show your work for each way you find x.

Type your response here:

How did you do? Check a box below.

Nailed It!—I included all of the same ideas as the model response on the Student Answer Sheet.

Halfway There—I included most of the ideas in the model response on the Student Answer Sheet.

Not Great—I did not include any of the ideas in the model response on the Student Answer Sheet.

4. Solving Right Triangles

In this activity, you will practice finding the side lengths and angle measures of various right triangles when given limited information. Go to the right triangle solver, and complete each step below. This activity will require careful planning of your calculations before you complete them. Be sure to think ahead.

• Generate a new problem for each of the problem types listed in the table. You’ll be choosing the unknown (what you’re solving for) and the method to generate the problem. If the problem onscreen does not fit the required type, click the New Problem button until you get a problem that does fit.

• The program will determine when you’ve properly constrained a problem and then give you a relevant equation. At this point, review the information that you’re given for the triangle and enter it in the appropriate column. Then enter the correct values into the equation onscreen by clicking on the correct parts of the triangle.

• Finally, determine your result. Solve for the unknown and enter your solution in the input window provided onscreen. Then paste a screenshot of your solution in the table.

• Every triangle in this section is a right triangle, so you can assume that one of the angles of each triangle equals 90° as part of your given information.

Type your response here:

|Solving for |Given |Method |Screenshot |

|acute angle | |sum of angles | |

|side | |Pythagorean Theorem | |

|acute angle | |definition of sine | |

|side | |definition of sine | |

|acute angle | |definition of cosine | |

|side | |definition of cosine | |

|acute angle | |definition of tangent | |

|side | |definition of tangent | |

How did you do? Check a box below.

Nailed It!—I included all of the same ideas as the model response on the Student Answer Sheet.

Halfway There—I included most of the ideas in the model response on the Student Answer Sheet.

Not Great—I did not include any of the ideas in the model response on the Student Answer Sheet.

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Lesson Activities

Geometry

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