Algebra 1 > Notes > Yorkcounty Final > Yorkcounty > Unit 8 - Polynomial > Square Roots And The Pythagorean Theorem
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Square Roots and the Pythagorean Theorem
The symbol for Square root is called a radical….
Inside the radical you put the number that you wish to take the square root of. For example:
25 The square root of 25 is 5. How does this work? What number times itself twice equals 25?? 5 times 5 equals 25.
What about the square root of 9?
9
What times itself two times equals 9?? 3 times 3.. so the square root of 9 is 3.
A number that has a square root which is a whole number is called a “perfect square.” 25 is one example because 25 = 5x5. Here is a table of some common perfect square numbers and their square roots. Perfect Square
Square Root
1
1
4
2
9
3
16
4
25
5
36
6
49
7
And so on…...
What about non perfect squares? We know that 4 is a perfect square and that 9 is also a perfect square. What about 5?? That is not a perfect square. We need to know how to express that as a decimal. In the SOL Test you will be required to give a decimal approximation to a non-perfect square root. Ex.
5
Non-Perfect Squares
5
What you will have to do here is use your graphing/scientific calculator. For a TI-83 you would use the following key sequence. To find 5 you would press the following key sequence… 1)
2nd
2)
x2
3) 5 ( or any number you want the square root of) 4)
enter
Try to do this now…...
Answer When you enter the square root of 5 in the calculator you should get an answer of 2.236067977… What you will be required to do now is round your answer to a certain decimal place. The tenths place is the number right after the decimal. After that is the hundredths and the thousands place. If you are asked to round to the tenths place you would look at the number, which is 2, and the number directly after that, which is 3. Since 3 is not greater than or equal to 5 you leave the 2 alone and your answer would be 2.2
Rounding Perhaps we got an answer of 2.36848 for a solution and we wanted to round to the tenths place. The number in the tenths place is 3 and the number after that is 6. Since 6 is greater than or equal to 5 we have to round the 3 up to a 4. Our answer would become 2.4 2.36848 rounded to the tenths place becomes 2.4 What happens if you round 2.36848 to the hundredths place? It becomes 2.37
Try These Find the square roots, round to the tenths place.
45
102
66
Solutions
45
102
66
= 6.70820 rounded to the tenths place is 6.7
= 10.099504 rounded to the tenths place is 10.1
= 8.124038 rounded to the tenths place is 8.1
Products of Square Roots Now we want to multiply two square roots together. Try this..
7⋅ 7 =7
Products of Square Roots
5⋅ 5 =5 7 ⋅ 6 = 6.4807
The distance, in miles, from an observation tower h feet above the ground to the horizon is d= 3h . How far a 2
distance, to the nearest tenth of a foot, can you see if the tower is 77 feet high?
Solution: we must substitute 77 for the height of the tower in the formula:
3 ⋅ 77 2
When this is calculated in the calculator you get 10.74709263 and must round it to the nearest tenth. 10.7 miles.
A right triangle is a triangle which has one right angle. The two sides which meet at the right angle are called the legs and the side which is opposite the right angle is called the hypotenuse.
hypotenuse leg
leg You can see that the hypotenuse is the longest side of the triangle.
Usually we label the legs a and b and the hypotenuse c. c a
b In geometry we learn that there is a special relationship between these three sides of the triangle. The sum of the squares of the lengths of the legs equals the square of the hypotenuse. This leads to the equation……
a +b = c 2
2
2
leg 2 + leg 2 = hypotenuse 2 This only works in right triangles….
We can use this theorem to check and see if sides of certain lengths will make a right triangle. For example, if you get a triangle with side lengths of 3cm, 4cm, and 5cm: Will it make a right triangle? Well if it did, then the side of 5 cm would have to be the hypotenuse, c, because the hypotenuse has to be the longest side. Let’s use the pythagorean theorem and substitute the numbers to see if we get a true statement. We can match the 3cm and 4cm with a and b in any order:
a +b = c 2
2
3 +4 =5 9 + 16 = 25 2
2
2
25 = 25
2 This is a true statement so the triangle is a right triangle.
If we substitute the given values from the problem we get... P=10 S=6 g We can set up the pythagorean theorem equation:
leg 2 + leg 2 = hypotenuse 2 g 2 + s2 = p 2 g 2 + 62 = 102 g 2 + 36 = 100 g 2 + 36 − 36 = 100 − 36
g 2 = 64 g 2 = 64
g=8
We must solve the equation using both old and new techniques to find the value of g.
Try This: Find the length of ‘g’ in the right triangle below:
13
12
g
Try This Solution: Using the pythagorean theorem we start with
leg 2 + leg 2 = hypotenuse 2
g + 12 = 13 2
2
2
g + 144 = 169 2
g 2 + 144 − 144 = 169 − 144
g = 25 g 2 = 25 2
g =5
13
12
g
The symbol for Square root is called a radical….
Inside the radical you put the number that you wish to take the square root of. For example:
25 The square root of 25 is 5. How does this work? What number times itself twice equals 25?? 5 times 5 equals 25.
What about the square root of 9?
9
What times itself two times equals 9?? 3 times 3.. so the square root of 9 is 3.
A number that has a square root which is a whole number is called a “perfect square.” 25 is one example because 25 = 5x5. Here is a table of some common perfect square numbers and their square roots. Perfect Square
Square Root
1
1
4
2
9
3
16
4
25
5
36
6
49
7
And so on…...
What about non perfect squares? We know that 4 is a perfect square and that 9 is also a perfect square. What about 5?? That is not a perfect square. We need to know how to express that as a decimal. In the SOL Test you will be required to give a decimal approximation to a non-perfect square root. Ex.
5
Non-Perfect Squares
5
What you will have to do here is use your graphing/scientific calculator. For a TI-83 you would use the following key sequence. To find 5 you would press the following key sequence… 1)
2nd
2)
x2
3) 5 ( or any number you want the square root of) 4)
enter
Try to do this now…...
Answer When you enter the square root of 5 in the calculator you should get an answer of 2.236067977… What you will be required to do now is round your answer to a certain decimal place. The tenths place is the number right after the decimal. After that is the hundredths and the thousands place. If you are asked to round to the tenths place you would look at the number, which is 2, and the number directly after that, which is 3. Since 3 is not greater than or equal to 5 you leave the 2 alone and your answer would be 2.2
Rounding Perhaps we got an answer of 2.36848 for a solution and we wanted to round to the tenths place. The number in the tenths place is 3 and the number after that is 6. Since 6 is greater than or equal to 5 we have to round the 3 up to a 4. Our answer would become 2.4 2.36848 rounded to the tenths place becomes 2.4 What happens if you round 2.36848 to the hundredths place? It becomes 2.37
Try These Find the square roots, round to the tenths place.
45
102
66
Solutions
45
102
66
= 6.70820 rounded to the tenths place is 6.7
= 10.099504 rounded to the tenths place is 10.1
= 8.124038 rounded to the tenths place is 8.1
Products of Square Roots Now we want to multiply two square roots together. Try this..
7⋅ 7 =7
Products of Square Roots
5⋅ 5 =5 7 ⋅ 6 = 6.4807
The distance, in miles, from an observation tower h feet above the ground to the horizon is d= 3h . How far a 2
distance, to the nearest tenth of a foot, can you see if the tower is 77 feet high?
Solution: we must substitute 77 for the height of the tower in the formula:
3 ⋅ 77 2
When this is calculated in the calculator you get 10.74709263 and must round it to the nearest tenth. 10.7 miles.
A right triangle is a triangle which has one right angle. The two sides which meet at the right angle are called the legs and the side which is opposite the right angle is called the hypotenuse.
hypotenuse leg
leg You can see that the hypotenuse is the longest side of the triangle.
Usually we label the legs a and b and the hypotenuse c. c a
b In geometry we learn that there is a special relationship between these three sides of the triangle. The sum of the squares of the lengths of the legs equals the square of the hypotenuse. This leads to the equation……
a +b = c 2
2
2
leg 2 + leg 2 = hypotenuse 2 This only works in right triangles….
We can use this theorem to check and see if sides of certain lengths will make a right triangle. For example, if you get a triangle with side lengths of 3cm, 4cm, and 5cm: Will it make a right triangle? Well if it did, then the side of 5 cm would have to be the hypotenuse, c, because the hypotenuse has to be the longest side. Let’s use the pythagorean theorem and substitute the numbers to see if we get a true statement. We can match the 3cm and 4cm with a and b in any order:
a +b = c 2
2
3 +4 =5 9 + 16 = 25 2
2
2
25 = 25
2 This is a true statement so the triangle is a right triangle.
If we substitute the given values from the problem we get... P=10 S=6 g We can set up the pythagorean theorem equation:
leg 2 + leg 2 = hypotenuse 2 g 2 + s2 = p 2 g 2 + 62 = 102 g 2 + 36 = 100 g 2 + 36 − 36 = 100 − 36
g 2 = 64 g 2 = 64
g=8
We must solve the equation using both old and new techniques to find the value of g.
Try This: Find the length of ‘g’ in the right triangle below:
13
12
g
Try This Solution: Using the pythagorean theorem we start with
leg 2 + leg 2 = hypotenuse 2
g + 12 = 13 2
2
2
g + 144 = 169 2
g 2 + 144 − 144 = 169 − 144
g = 25 g 2 = 25 2
g =5
13
12
g
