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Proofs and Use of the Pythagorean Theorem

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Derive and Use the Pythagorean Theorem
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Have you ever painted something taller than you are? Take a look at this dilemma.

While Mr. Kennedy’s class was working on the garden, the students in Ms. Richardson’s class decided to paint the equipment shed where all of the sports equipment was kept.

“That thing hasn’t been painted in decades,” Karen said in the first meeting.

“I agree, it does look awful,” Cameron added.

“Well, I don’t know about decades, but it does need to be painted, so that is what we are going to do. Now to work on this project, we will need to choose a ladder that will reach up high enough. How can we figure this out?” Ms. Richardson asked.

The class was silent.

“I have an idea,” Veria said smiling. “It has to do with triangles. We need to figure out the height of the ladder, compared to the height of the building.”

“Yes, but don’t forget that the ladder reaches out from the building, not up against it. So we have to consider that measurement too,” Aran chimed in.

“What can we use to figure that out?” Karen asked.

Once again, the group was silent.

A mathematical formula involving the Pythagorean theorem is needed to solve this problem. The students have their work cut out for them. So do you. Pay attention to this Concept so that you can explain the formula that they will need and why they will need it.

Guidance

You have probably already studied many different types of triangles.

Acute triangles have angles that are all less than 90^{\circ} .

Obtuse triangles have one angle that is between 90^{\circ} and 180^{\circ} .

Right triangles have one angle that measures exactly 90^{\circ} —in other words, it has one right angle.

This Concept focuses entirely on properties specific to right triangles. While all of the equations and strategies you are about to learn are helpful, they apply only to right triangles – they will not work with acute or obtuse triangles.

To begin, let’s look at the parts of a right triangle.

The legs are the two sides of the triangle that are labeled a and b . The hypotenuse is the longest side of a right triangle and it is labeled c . There is a special relationship between the legs of a right triangle and the hypotenuse of a right triangle.

One of the special characteristics of right triangles is described by the Pythagorean Theorem , thought to have been developed around 500 B.C.E. It states that the squared value of the hypotenuse will equal the sum of the squares of the two legs. In the triangle above, the sum of the squares of the legs is a^2 + b^2 and the square of the hypotenuse is c^2 . So, the Pythagorean theorem is commonly represented as a^2 + b^2 = c^2 where a and b are the legs of the right triangle and c is the hypotenuse.

The Pythagorean Theorem is known as a^2 + b^2 = c^2 .

You may be asking yourself why that is the case. Well, we can think about the Pythagorean Theorem in terms of a square. We know that there is a relationship between a square and a right triangle. We can divide a square with a diagonal and because a square has four right angles, the diagonal will divide the square into two right triangles. Now because a right triangle comes from the square, the sides will also be related to the square. This is where the Pythagorean Theorem comes from.

Let’s look at one.

Use the measures of the triangle below to test the Pythagorean theorem.

The legs of the triangle above are 3 inches and 4 inches. The hypotenuse is 5 inches. So, a = 3 , b = 4 , and c = 5 . We can test the formula to see if this is true.

a^2 + b^2 &= c^2\\3^2 + 4^2 &= 5^2\\(3 \times 3) + (4 \times 4) &= (5 \times 5)\\9 + 16 &= 25\\25 &= 25

Since both sides of the equation equal 25, the equation is true. Therefore, the Pythagorean theorem worked on this right triangle.

This combination of numbers (3, 4, 5) is referred to as a Pythagorean triple . In other words, these three numbers work together to make the Pythagorean Theorem true.

Now that you have learned how to derive and execute the Pythagorean Theorem, there are many different ways to apply it. Any time you have two out of three sides in a right triangle, you can find the third using the equation a^2 + b^2 = c^2 , where a and b are the lengths of the legs of the triangle and c is the length of the hypotenuse.

When applying the Pythagorean Theorem, be sure to use exponents and square roots accurately.

What is the length of b in the triangle below?

Use the Pythagorean Theorem to identify the length of the missing leg, b . Be sure to simplify the exponents and roots carefully. Also remember to use inverse operations to solve the equation properly.

a^2 + b^2 = c^2, where a = 6 and c = 10

6^2 + b^2 &= 10^2\\36 + b^2 &= 100\\36 + b^2 - 36 &= 100-36\\b^2 &= 64\\\sqrt{b^2} &= \sqrt{64}\\b &= 8

The length of the missing side is 8 inches.

You already know about the Pythagorean triple 3:4:5. Notice that this triangle is proportional to that ratio. If you divide the lengths of the triangle in the example by two, you find the same proportion—3:4:5. Whenever you find a Pythagorean triple, you can apply those ratios with greater factors as well. So, 6, 8, 10 is another Pythagorean triple.

Notice that as long as you use the Pythagorean Theorem you can figure out the missing length of any of the three sides of a right triangle.

Find the missing side length for each right triangle.

Example A

9,12

Solution:  15

Example B

15,20,

Solution:  25

Example C

21,28,

Solution:  35

Now let's go back to the dilemma from the beginning of the Concept.

The students will need to use the Pythagorean Theorem to figure out this problem.

a^2 + b^2 = c^2

Why? They will need to use the Pythagorean Theorem because the ladder against the shed forms a right triangle with the ground. The shed and the distance that the ladder is placed from the shed form the sides a and b of the right triangle. The ladder is the c side of the triangle.

Look at the diagram below.

Vocabulary

Right Triangle
one angle is equal to 90^{\circ} .
Legs
the two shorter sides of a right triangle.
Hypotenuse
the longest side of a right triangle.
Pythagorean Theorem
a^2 + b^2 = c^2
Pythagorean Triple
values that work perfectly in the Pythagorean Theorem. The ratio always simplifies to 3:4:5.

Guided Practice

Here is one for you to try on your own.

Find the length of the missing side in the triangle below.

Solution

Use the Pythagorean Theorem to identify the length of the missing hypotenuse. Be sure to simplify the exponents and roots carefully. Also remember to use inverse operations to solve the equation properly.

a^2 + b^2 = c^2, where a = 5 and b = 12

5^2 + 12^2 &= c^2\\25 + 144 &= c^2\\169 &= c^2\\\sqrt{169} &= \sqrt{c^2}\\13 &= c

The length of the missing side is 13 centimeters.

Video Review

Khan Academy Introduction to the Pythagorean Theorem

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Directions: Use the Pythagorean Theorem to find the missing dimensions of right triangles.

  1. a=3, \ b=4, \ c=?
  2. a=6, \ b=8, \ c=?
  3. a=9, \ b=12, \ c= ?
  4. a=27, \ b=36, \ c= ?
  5. a=15, \ b=20, \ c= ?
  6. a=18, \ b=24, \ c= ?
  7. a= ?, \ b=16, \ c= 20
  8. a= ?, \ b=28, \ c=35
  9. a=30, \ b= ?, \ c=50
  10. a=33, \ b= ?, \ c=55
  11. a=1.5, \ b= ?, \ c=2.5
  12. a=36, \ b= ?, \ c=60

Directions: Answer the following questions True or False.

  1. The Pythagorean Theorem will work for any triangle.
  2. The longest side of a right triangle is called the hypotenuse.
  3. A Pythagorean Triple can only be found in a right triangle.

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