Transcript Pythagorean Theorem
Pythagorean Theorem
•
MACC.8.G.2.7 -
Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
Created by: Matthew Funke 8 th Grade Math Teacher Central Middle School West Melbourne, FL
Bell Ringer
• • Solve for x x 2 +7=43 64+x 2 =164
Ans: x = Ans: x = ±6 ±10
Evaluate for a = 12, b = 5, c = 13 3. a 2 + b 2 4. c 2 – b 2
Ans: 169 Ans: 144
Today’s Objective
No need For notes On this slide • We are going to learn more about the Pythagorean Theorem.
• Today, we are going to learn how to use the Pythagorean Theorem to solve for a missing length of a right triangle.
Pythagorean Theorem
No need For notes On this slide • What is the Pythagorean Theorem in symbol form?
a 2 + b 2 = c 2 • Which of these variables represent the hypotenuse?
c • Once you have figured out which is C, does it matter which leg is A and which is B?
no
Steps to Solve for a missing Side of a triangle using the Pythagorean Theorem
TAKE NOTES
Step 1: Write the formula Step 2: Substitute known values for the variables.
Step 3: Solve the equation for the missing variable.
Find x
Example 1
8 ft x TAKE NOTES
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Step 1:
Write the formula
15 ft a 2 + b 2 = c 2
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Step 2:
Substitute known values
8 2 + 15 2 = c 2
Which number goes where? You need to identify the hypotenuse. It’s the one opposite of the right angle.
The hypotenuse is always going to be the c in the formula. Since we do not know the value of c, it stays as c in the formula.
Does it matter whether we use a = 8 or 15? No. Let’s use a = 8 and b = 15.
Find x
Example 1
8 ft x TAKE NOTES
•
Step 1:
Write the formula
15 ft a 2 + b 2 = c 2
• •
Step 2:
Substitute known values
8 2 + 15 2 = c 2 Step 3:
Solve for the missing variable, in this case c.
We are not done yet… We have found c 2 , but not just plain c.
We were told to solve for x, not c. So we should replace the c with an x. x = 17 64 + 225 = c 2 289 = c 2 289 = c 2 17 = c
You try this one in your notes.
TAKE NOTES x Find x 5 ft 12 ft 5 2 + 12 2 = x 2 25 + 144 = x 2 169 = x 2
• Answer:
x = 13
Example #2
TAKE NOTES
• •
14 in x Find x.
Round to the nearest hundredth.
6 in Step 1:
Write out the formula
a 2 + b 2 = c 2 a 2 Step 2:
Substitute known values Which number goes where? This time we are given the hypotenuse. So, c = 14
+ 6 2 = 14 2
Does it matter whether we use a = 6 or b = 6? No Let’s use b = 6.
Example #2
TAKE NOTES 14 in x Find x.
Round to the nearest hundredth.
6 in
• • •
Step 1: Step 2:
Write out the formula Substitute known values
a a 2 2 + b + 6 2 2 = c 2 = 14 2 Step 3:
Solve for the missing variable, in this case a.
Can we just add the two numbers and do the square root? No, they are not on
a 2 + 36 = 196 – 36 – 36
the same side of the equals sign.
a 2 = 160 x = 12.65
a 2 = 160 a = 12.64911
What is the difference between the 2 examples?
No need For notes On this slide • Both have you squaring the given sides.
• Both have you using the square root at the end.
• The only difference is in the middle.
– Example 1 has you adding the numbers – Example 2 has you subtracting the smaller from the larger.
What does this mean?
• When you have two sides of a right triangle, you can find the third using the Pythagorean Theorem.
• Square both of the measurements you have.
• Add or subtract the two numbers depending on whether or not you have the hypotenuse. (Subtract if you have it, add if you don’t) • Find the square root of the result and you have your missing side!
Try this one in your notes…
x 15 20 Solve for x. Round your answer to the nearest hundredth if necessary.
Answer: 25
Try this one in your notes…
7 12 x Solve for x. Round your answer to the nearest hundredth if necessary.
Answer: 13.89
Try this one in your notes…
5 x 3 Solve for x. Round your answer to the nearest hundredth if necessary.
Answer: 4
Try this one in your notes…
30 7 Solve for x. x Round your answer to the nearest hundredth if necessary.
Answer: 30.81
Try this one in your notes…
b a c If the hypotenuse of this triangle is 10 and
a
is 6 which equation would you use to find b?
a) 6 + b = 10 b) 36 + b = 100 c) b 2 – 36 = 100 d) 36 + b 2 = 100