Math 11 Apprenticeship & Workplace Pythagorean Theorem
Course Materials Booklet
Instructions for SA Completion
1- Take notes on these Pythagorean Theorem Course Materials then do
and check the associated practice questions for an explanation on how to
do the Pythagorean Theorem Substantive Assignment questions correctly.
2- Go to the Online Unit 1 Course Materials website (below) for an explanation
on how to do the Trigonometry Ratio Substantive Assignment questions
correctly.
http://contentconnections.ca/sides/caaAW11/Lesson1/Lesson1ControllerStream.html
http://www.surreyconnect.sd36.bc.ca/Secondary/Courses/grade11/Pages/Math11-Apprenticeship%20and%20workplace.aspx
http://contentconnections.ca/sides/caaAW11/Lesson2/Lesson2ControllerStream.html
http://contentconnections.ca/sides/caaAW11/Lesson3/Lesson3ControllerStream.html
3- Take notes on the video Lessons 1&2 using the note taking supplements.
4- Complete the practice assignments from Lessons 1&2.
5- Check the practice assignment with the answers provided.
6- Check the video solutions to the questions you did not do correctly.
7- Seek help with questions the video solutions do not fully explain to you.
8- Complete the Substantive Assignment questions showing all work to earn
full marks for each question.
9- Seek help if you are confused with any questions.
10- When you are happy with your final product please scan and submit your
Substantive Assignment with your registration package.
11-While you are waiting for your registration to be processed, please complete
all the Online Unit 1 Course Materials Lessons in preparation for your Unit 1
Exam.
Page 1 of 20
Math 11 Apprenticeship & Workplace Pythagorean Theorem
Course Materials Booklet
Pythagorean Theorem Course Materials
In mathematics, the Pythagorean Theorem is a relation among the three sides of a right
triangle (right-angled triangle).
The theorem can be written as an equation (or formula) relating the lengths of the
sides a, b and c, often called the Pythagorean equation:
The longest side of a right-angled triangle is found across from the right angle and is called the
hypotenuse. The hypotenuse is represented by the letter “c” in the Pythagorean formula. The
legs of the triangle are the two shorter sides. The legs meet perpendicularly to create a 90 degree
angle. The right angle is represented by a square in the triangle and the legs of the triangle are
represented by the letters “a” and “b” in the formula. It doesn’t matter which leg is labeled “a”
or “b”. Please note the sides “a”, “b” and “c” can be found across from the respective angles
“A”, “B”, and “C”.
If the length of both “a” and “b” are known, then “c” can be calculated as follows:
You can find the length of a hypotenuse by using the formula:
This example asks us to find the approximate length of the hypotenuse.
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Math 11 Apprenticeship & Workplace Pythagorean Theorem
Course Materials Booklet
Try the question above and check the solution on the next page.
Hint use:
or this algebraically manipulated form of the same equation:
√
√
√
This form of the formula is the same formula written in a different algebraic way:
Page 3 of 20
Math 11 Apprenticeship & Workplace Pythagorean Theorem
Course Materials Booklet
Finding the length of the legs in a right triangle requires some algebraic manipulation of the
Pythagorean Formula.
If the length of hypotenuse “c” and one leg (“a” or “b”) are known, then the length of the other
leg can be calculated with the following equations:
or
Page 4 of 20
Math 11 Apprenticeship & Workplace Pythagorean Theorem
Course Materials Booklet
“Solve the triangle” means to find all missing sides and angles possible. In this case we can only solve for
the missing side:
Please note in the previous and next solutions you may isolate the variable (unknown letter)
before you insert the known values or after you insert the known values. Both methods lead to
the correct answer.
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Math 11 Apprenticeship & Workplace Pythagorean Theorem
Course Materials Booklet
We need to be able to find exact answers as well as the approximate decimal answers we’ve been
calculating in the previous examples. An exact answer is a real number not approximated by
rounding the decimal portion of the answer.
Example showing how to find the exact value (simplified using a square root symbol):
1) Find the exact value for the missing side:
6m
a
4m
Page 6 of 20
Math 11 Apprenticeship & Workplace Pythagorean Theorem
Course Materials Booklet
Solution
1)
Subtract 16 from both sides to help isolate the unknown “a”
Square root both sides to isolate “a”
a=√
At this point you can calculate an approximation for this exact expression with your calculator to get
an answer of 4.47cm however this question has asked for an exact answer.
Therefore we will try to find a perfect root that is a factor of 20. Perfect roots are found by
multiplying an integer by itself. For example 2 multiplied by 2 is 4. Four is a perfect root because the
√ = 2. Other perfect roots include: 9, 16, 25, 36, 49, etc…
In this case 4 is the largest perfect root that is a factor of 20. That allows us to write the answer in the
following simplified form:
a=√
a= √
a=2 √
√
√
cm
Page 7 of 20
Math 11 Apprenticeship & Workplace Pythagorean Theorem
Course Materials Booklet
We have shown that if the lengths of any two sides are known the length of the third side can be
found through the use of Pythagorean equation. If a triangle has three known sides we can also
use the Pythagorean equation to prove if it is, or is not, a right-angled triangle:
Determine if these Triangles are right-angled ones or not:
1a)
11m
11m
11√ m
Solution:
Please note this triangle may look like it has a right angle in it but because there is no square symbol
inside the triangle we cannot assume this triangle has a right angle.
We can use the Pythagorean equation to determine if the triangle is right-angled
√
121 + 121 = (121) (2)
242 = 242
We used the Pythagorean equation calculation to determine that both sides of the equation are equal.
This is a true statement. That means all the conditions that allow the Pythagorean theorem to be
true, are true. One of the key features of using the Pythagorean theorem is that you are using a rightangled triangle to do your calculations.
Therefore this is a right-angled triangle.
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Math 11 Apprenticeship & Workplace Pythagorean Theorem
Course Materials Booklet
1b)
12m
11m
16m
Solution:
We can use the Pythagorean equation to determine if the triangle is right-angled:
121 + 144 = 256
265 ≠ 256
We used the Pythagorean equation calculation to determine that both sides of the equation are not
equal. This means the equation creates a false statement. That means all the conditions that allow
the Pythagorean Theorem to be true are NOT true and one of the key features of using
Pythagorean’s theorem is that you are using a right-angled triangle to do your calculations.
Therefore this is a not a right-angled triangle.
Page 9 of 20
Math 11 Apprenticeship & Workplace Pythagorean Theorem
Course Materials Booklet
Practice Questions
a) ________________________
b) ________________________
Page 10 of 20
Math 11 Apprenticeship & Workplace Pythagorean Theorem
Course Materials Booklet
c) ________________________
2. In each diagram, calculate the indicated side exactly using Pythagorean Theorem:
a)
5cm
3cm
b
a) ________________________
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Math 11 Apprenticeship & Workplace Pythagorean Theorem
Course Materials Booklet
b)
y
7km
3km
b) ________________________
3. Prove which triangle is right-angled and which is not.
Triangle B _________________ Triangle C_______________
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Math 11 Apprenticeship & Workplace Pythagorean Theorem
Course Materials Booklet
4) Find and prove which Triangles are right and which are not:
a)
A - ______________
B - _______________
C - _______________
D - _______________
Page 13 of 20
Math 11 Apprenticeship & Workplace Pythagorean Theorem
Course Materials Booklet
Practice Questions Answers
1a) h = approximately 4.70 cm
1b) r = approximately 11.32 m
1c) f = approximately 11.43 in
2a) b = 4 cm
2b) y = 2√
3)
km
Triangle B is not Right triangle and Triangle C is a right triangle.
4) Triangles A is a right triangle and Triangles B, C and D are not right triangles
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Math 11 Apprenticeship & Workplace Pythagorean Theorem
Course Materials Booklet
Practice Questions Full Solutions
a)
22.1
√
=h
4.701063709 =h
a)
h = approximately 4.70 cm
60.84
128.08
√
=r
11.31724348 = r
b) r = approximately 11.32 m
Page 15 of 20
Math 11 Apprenticeship & Workplace Pythagorean Theorem
Course Materials Booklet
42.25
130.61
√
=f
11.42847321 = f
c) f = approximately 11.43 in
2. In each diagram, calculate the indicated side exactly using Pythagorean Theorem:
a)
5cm
3cm
b
9
= 25
= 25 - 9
b= √
b=4
a)
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b = 4 cm
Math 11 Apprenticeship & Workplace Pythagorean Theorem
Course Materials Booklet
b)
y
7km
9
3km
= 49
= 49 - 9
y= √
y= √
√
y= 2(√
)
y = 2√
3. Prove which triangle is right-angled and which is not.
25 + 144 = 196
169 = 196
This is a false statement
Therefore B is not a right-angled triangle.
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km
Math 11 Apprenticeship & Workplace Pythagorean Theorem
Course Materials Booklet
36 + 64 = 100
100 = 100
This is a true statement
Therefore C is a right-angled triangle.
4) Find and prove which Triangles are right and which are not:
A – right triangle
A)
B)
51.84 + 29.16 = 81
36 + 64 = 81
81 = 81
100 = 81
This is a true statement
This is a false statement
C)
D)
12.25 + 20.25 = 30.25
25 + 100 = 225
32.5 = 30.25
125 = 225
This is a false statement
This is a false statement
B – not a right triangle
C - not a right triangle
D - not a right triangle
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