Download Solving Problems with Right Triangles The Lesson Activities will

Geometry
Lesson Activities
Student Answer Sheet
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.
_________________________________________________________________________
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 nonright 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 AC, and label the intersection of the
perpendicular line and AC point D. Measure and record BD and m BDA . 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.)
Sample answer:
BD = 8
m BDA = 90°
∆ABD and ∆BCD are both right triangles, each with two known side lengths.
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© 2013 EDMENTUM, INC.
b. Use the Pythagorean Theorem to find AD and DC. Then find AC using addition. Show
your calculations.
Sample answer:
AB 2  BD 2 =
10 2  8 2 =
36 = 6
DC = BC 2  BD2 =
AC = AD + DC = 21
17 2  8 2 =
225 = 15
AD =
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.
Sample answer:
Answers will vary depending on the given side lengths.
AB
AD
BD
AB2
AD2 + BD2
8
6.44
4.75
64
64.04
12
9.34
7.55
144
144.24
14
8.78
10.91
196
196.11
16
11.94
10.62
BC
DC
256
BC2
BD
255.35
DC2
+ BD2
15.32
14.56
4.75
234.70
234.56
13.89
11.66
7.55
192.93
192.96
16.38
12.22
10.91
268.30
268.36
13.95
9.06
10.62
194.60
194.87
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 A as the right angle. Fill in
the fraction forms of sin B, cos B, tan B and then the associated measure of B 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 C using sin-1, cos-1, and
tan-1 rounded to three decimal places.
Sample answer:
Answers will vary depending on the given side lengths. This set of values corresponds
to AC = 150.
2
p = sin B
 AC 


 BC 
q = cos B
 AB 


 BC 
r = tan B
 AC 


 AB 
150
183.12
105.03
183.12
150
105.03
x = sin C
 AB 


 BC 
y = cos C
 AC 


 BC 
z = tan C
 AB 


 AC 
105.03
183.12
150
183.12
105.03
150
m B =
sin-1 p
54.998°
m C =
sin-1 x
34.999°
m B =
cos-1 q
55.001°
m C =
cos-1 y
35.002°
m B =
tan-1 r
55.000°
m C =
tan-1 z
34.999°
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°.
Sample answer:
The angle measurements obtained in each part of the table are equal within the limits
of accuracy despite the different inverse functions used. The sum of interior angles is
A  B  C = 90° + 55° + 35° = 180°.
c. Click the refresh button until you get a right triangle with B as the right angle, and fill
in the table below as you did in part a.
Sample answer:
Answers will vary depending on the given side lengths. This set of values corresponds
to AC = 50.
p = sin A
q = cos A
r = tan A
 BC 
 BC 
 AB 
m A =
m A =
m A =






 AC 
 AC 
 AB 
sin-1 p
cos-1 q
tan-1 r
40.96
50
28.68
50
40.96
28.68
x = sin C
 AB 


 AC 
y = cos C
 BC 


 AC 
z = tan C
 AB 


 BC 
28.68
50
40.96
50
28.68
40.96
55.005
m C =
sin-1 x
35.002
3
54.998
m C =
cos-1 y
34.995
55.000
m C =
tan-1 z
35.000
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.
Sample answer:
The angle measurements obtained in each part of the table are equal within the limits
of accuracy despite the different inverse functions used. The sum of interior angles is
A  B  C = 55°+ 90°+ 35° = 180°.
e. Click the refresh button until you get a right triangle with C as the right angle, and fill
in the table below as you did in parts a and c.
Sample answer:
Answers will vary depending on the given side lengths. This set of values corresponds
to AC = 50.
p = sin A q = cos A
r = tan A
 BC 
 AC 
 BC 
m A =
m A =
m A =






-1
-1
AB
AC
AB






sin p
cos q
tan-1 r
28.87
57.74
50
57.74
28.87
50
x = sin B
 AC 


 AB 
y = cos B
 BC 


 AB 
z = tan B
 AC 


 BC 
50
57.74
28.87
57.74
50
28.87
30.000
m B =
sin-1 x
59.991
30.009
m B =
cos-1 y
60.000
30.002
m B =
tan-1 z
59.998
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°.
Sample answer:
The angle measurements obtained are equal within the limits of accuracy despite the
different inverse functions used. The sum of interior angles is A  B  C = 60°+
30°+ 90°= 180°.
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?
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Sample answer:
AD = 120 cm
1
1
AC = AD = (120) = 60 cm
2
2
1
. Since the two right triangles
2
1
that make up the kite are congruent, then sinCBD =
as well. Using the definition of
2
sine, what is AB? Show your work.
b. The instructions in the kite book state that sinABC =
Sample answer:
1
sinABC =
2
AC 1
=
AB 2
AC = 60 cm
60
1
=
AB 2
AB = 120 cm
c. Using inverse trigonometric functions, what is m ABC ?
Sample answer:
1
m ABC = sin-1
= 30°
2
d. Using m ABC , find m CAB . Show your work.
Sample answer:
m CAB = 180° – 90° – 30° = 60°
e. Let x represent BC. Using either ABC or CAB 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.
Sample answer:
x
cos 30° =
120
120 cos 30° = x
x ≈ 103.923 cm
x
60
60 tan 60° = x
x ≈ 103.923 cm
x
120
120 sin 60° = x
x ≈ 103.923 cm
sin 60° =
tan 60° =
5
tan 30° =
x=
60
x
60
≈ 103.923 cm
tan30
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.
Sample answer:
Solving for
Given
Method
Screenshot
acute angle
one acute angle
sum of angles
Answers will vary.
side
two sides
Pythagorean Theorem
Answers will vary.
acute angle
hypotenuse and
length of leg
opposite the
unknown angle
definition of sine
Answers will vary.
side
one acute angle
definition of sine
and length of leg
opposite the angle
Answers will vary.
acute angle
hypotenuse and
length of leg
adjacent to the
unknown angle
definition of cosine
Answers will vary.
side
one acute angle
and length of leg
adjacent to the
angle
definition of cosine
Answers will vary.
acute angle
lengths of legs
opposite and
adjacent to the
unknown angle
definition of tangent
Answers will vary.
side
one acute angle
definition of tangent
and length of leg
opposite the angle
Answers will vary.
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