Simplifying Radicals and the Pythagorean Theorem

Pythagorean Theorem

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The Pythagorean Theorem is a mathematical rule that applies to right triangles.

Let's check out how it works!

Don't quite believe me that this is always true? Thinking to yourself "Ms. Close, I am going to need a little more evidence"?

Check out this second way to look at it.

​Watch as the water in the two smaller squares exactly fills the largest square...

they must be equal!

The Pythagorean Theorem says that the sum of the squares built on the two shorter sides of a right triangle will equal the square built on the longest side.

We shorten that by calling the sides a, b and c, and using the formula a² + b² = c² .

But who's who? Well, right triangles have two legs and a hypotenuse.

The legs ( "a" and "b") are always the two shorter sides.

The hypotenuse ("c") is always the longest side and across from the right angle.

Multiple Choice

The Pythagorean Theorem ONLY works on which triangle? 

• obtuse

• scalene

• isosceles

• right

Multiple Choice

What is the side of a right triangle across from the right angle called?

• square

• arm

• leg

• hypotenuse

Multiple Choice

If a right triangle has side lengths 6, 6.7 and 3, which side is the hypotenuse?

• 6

• 6.7

• 3

• 15.7

• None of them

So how do we use this Theorem? In two ways:

• If we know two of the three lengths of a right triangle, we can find the missing side length

• If we know all three sides of a triangle, we can prove if it is a right triangle

But how does knowing the area of a square help us find the length of a side?

Now it's your turn to try a few!

Get some paper to work on....

like your spiral notebook!

Find the length of the missing side using

a² + b² = c²

When you have your answer, move to the next slide.

Find the length of the missing side using

a² + b² = c²

When you have your answer, move to the next slide.

Multiple Choice

Which equation can be used to solve for "x"?

• 15² – x² = 21²

• x² – 15² = 21²

• 15² + 21² = x²

• 21² – 15² = x²

Multiple Choice

^{Find the missing side}

• 11

• 61

• $$\sqrt{61}$$

• $$\sqrt{11}$$

Multiple Choice

_{^{Find the missing side}}

• 29.2

• 20

• 400

• 10

Multiple Choice

Determine the length of the missing side.

• 7

• 8

• 17

• 23

Multiple Choice

_{^{Find the missing side}}

• 9

• 41

• 6.4

• 3

Multiple Choice

_{^{Find the missing side}}

• 36
• 76.8
• 12
• 40

Multiple Choice

Solve for c

• 7

• 9

• 4

• 13

So you know how to use the Pythagorean Theorem to find a missing side? Great!

But you also said I could prove it was a right triangle if I know all the sides...

Multiple Choice

Would a triangle with these side lengths be a right triangle?
4, 5, 6

• Yes

• No

Multiple Choice

A triangle has sides of a = 2.1, b = 7.2, c = 7.5

Is this a right triangle?

• yes

• no

Multiple Choice

Which set of sides would make a right triangle?

• 4,5,6

• 8,10,12

• 5,12,13

• 5,10,12

So scaling up a known Pythagorean Triple can give you infinite possibilities for more of them.

And a few more with a context...

Helpful Hint: Draw a picture!

Multiple Choice

Casey travels from her house directly west to the bank and then directly north from the bank to the mall. She then travels home on the road connecting the mall and her house. What is her total distance traveled?

• 17 miles

• 32 miles

• 37 miles

• 40 miles

Multiple Choice

A piece of paper that Brittany has is 11 inches tall and 8 inches wide. She draws a straight line diagonally across the paper. How long is the line she drew?

• 7.5 in

• 12.4 in

• 13.6 in 

• 14.3 in

Multiple Choice

A baseball "diamond" is actually a square with sides of 90 feet. If a runner tries to steal second base, how far must the catcher, at home plate, throw to get the runner "out"? (Distance across the diagonal)

• 180 feet

• 90 feet

• 13.4 feet

• 127.3 feet

Multiple Choice

How far from the base of a house do you need to place a 15' ladder so that is exactly reaches the top of a 12' wall?

• 9 feet

• 19.2 feet

• 81 feet

• 369 feet

Multiple Choice

An airplane flies 56 miles due north and then 33 miles due east. How many miles is the plane from its starting point?

• 65 miles

• 89 miles

• 45.4 miles

• We don't have enough information.

Multiple Choice

The size of a television screen is measured by the length of the diagonal of the screen. What size is a television screen that is 3 feet tall and 6 feet wide?

• 18

• 6.7

• 5

• 3

• 4.5

The End