Instructions for SA Completion

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1 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 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

2 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. Page 2 of 20

3 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

4 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

5 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. Page 5 of 20

6 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: a 4m 6m Page 6 of 20

7 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

8 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) (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. Page 8 of 20

9 1b) 12m 11m 16m Solution: We can use the Pythagorean equation to determine if the triangle is right-angled: = 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

10 Practice Questions a) b) Page 10 of 20

11 c) 2. In each diagram, calculate the indicated side exactly using Pythagorean Theorem: a) 3cm 5cm b a) Page 11 of 20

12 b) y 7km 3km b) 3. Prove which triangle is right-angled and which is not. Triangle B Triangle C Page 12 of 20

13 4) Find and prove which Triangles are right and which are not: a) A - B - C - D - Page 13 of 20

14 Practice Questions Answers 1a) h = approximately 4.70 cm 1b) r = approximately m 1c) f = approximately in 2a) b = 4 cm 2b) y = 2 km 3) 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 Page 14 of 20

15 Practice Questions Full Solutions a) 22.1 = h =h a) h = approximately 4.70 cm = r = r b) r = approximately m Page 15 of 20

16 = f = f c) f = approximately in 2. In each diagram, calculate the indicated side exactly using Pythagorean Theorem: a) 3cm 5cm b 9 = 25 = 25-9 b = b = 4 a) b = 4 cm Page 16 of 20

17 b) y 7km 3km 9 = 49 y = = 49-9 y = y= 2( ) y = 2 km 3. Prove which triangle is right-angled and which is not = = 196 This is a false statement Therefore B is not a right-angled triangle. Page 17 of 20

18 = = 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 B not a right triangle C - not a right triangle D - not a right triangle A) = = 81 This is a true statement B) = = 81 This is a false statement C) D) = = This is a false statement Page 18 of = = 225 This is a false statement

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