NGSSSMA.8.G.2.4

The student will be able to:

Validate and apply Pythagorean Theorem to find distances in real world situations or between points in the coordinate plane.

CCSS8.G Understand and apply the

Pythagorean Theorem.

6. Explain a proof of the Pythagorean Theorem and its converse. 7. Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. 8. Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

The Pythagorean Theorem in HS:

The Pythagorean Theorem underlies several formulas and identities that are memorized

by high school students. Related formulas include

•The Distance formula

•The Law of Cosines

•The equation of a Circle

•Some trigonometric identities.

Often, students memorize these formulas in isolation, without being aware of their connection to the Pythagorean Theorem.

What is a right triangle?

• It is a triangle which has an angle that is 90 degrees.

• The two sides that make up the right angle are called legs.

• The side opposite the right angle is the hypotenuse.

leg

leg

hypotenuse

right angle

The Pythagorean Theorem

In a right triangle, if a and b are the measures of the legs and c is the

hypotenuse, then

a2 + b2 = c2.Note: The hypotenuse, c, is always

the longest side.

a2 + b2 = c2

http://www.pbs.org/wgbh/nova/proof/puzzle/theorem.html

The image is the logo

from the Institute for

Mathematics & Education.

It provides us with an

elegant geometric “proof”

of the Pythagorean

Theorem.

Activity: How does this illustration prove the Pythagorean Theorem?

Proof of the Pythagorean Theorem

Given the red right

triangle, prove that the

area of the square of the

hypotenuse is equal to

the sum of the areas of

the squares of the two

legs.

The figure is formed from two large adjacent squares.

Each large square contains four congruent right triangles, one of which is colored red.

Proof of the Pythagorean Theorem

The left square contains

two smaller squares.

The smallest square is

the result of the shorter

leg of the red right triangle.

The larger square is the result of the longer leg of the red right triangle.

The largest square at the right is the result of the hypotenuse of the red triangle.

Proof of the Pythagorean Theorem

Since both large squares

are equal, we can

subtract the four right

triangles from each

large square and still

have equal areas.

On the left are the squares of the two legs of the red right triangle. On the right is the square of the hypotenuse.

Therefore, in a right triangle, the sum of the squares of the two legs is equal to the square of the hypotenuse.

Proof of the Pythagorean Theorem

1. Find the length of the hypotenuse

122 + 162 = c2

144 + 256 = c2

400 = c2

Take the square root of both sides.

2400 c

12 in

16 in

The hypotenuse is 20 inches long.

52 + 72 = c2

25 + 49 = c2

74 = c2

Take the square root of both sides.

2. Find the length of the hypotenuse

274 c

5 cm

7 cm

The hypotenuse is about 8.6 cm long.

3. Find the length of the hypotenuse given that the legs of a right triangle

are 6 ft and 12 ft.

1. 180 ft.

2. 324 ft.

3. 13.42 ft.

4. 18 ft.

4. Find the length of the missing leg.

42 + b2 = 102

16 + b2 = 100-16 -16

b2 = 842 84b

4 cm 10 cm

The leg is about 9.2 cm long.

a2 + 122 = 132

a2 + 144 = 169 -144 -144

a2 = 25

5. Find the length of the missing leg.

12 in

13 in

The leg is 5 inches long.

6. Find the length of the missing side of a right triangle if one leg is 4

ft and the hypotenuse is 8 ft.

1. 24 ft.

2. 4 ft.

3. 6.9 ft.

4. 8.9 ft.

Application of Pythagorean Theorem

• The screen aspect ratio, or the ratio of the width to the length of a HDTV is 16:9. The size of a television is

given by the diagonal distance across the screen. If an HDTV is 41 inches wide, what is its diagonal screen size?

• What are the dimensions of a 65 inch HDTV?

Application of Pythagorean Theorem

• A baseball diamond is a square with 90-foot sides. What is the approximate distance the catcher must throw from home to second base?

The Converse of the Pythagorean Theorem

A common application of the converse of the

Pythagorean Theorem is used by carpenters to

make sure a corner that they are constructing forms

a right angle. Here are the steps:

1.Starting at the corner, measure 3 units along

one direction and make a mark.

2. Measure 4 units along the other direction and make a mark.

3. Measure the distance between the marks.

4. If the length is equal to 5 units, then the corner forms a right angle (90°)

If the length is less than 5 units, then the corner is less than 90°

If the length is greater than 5 units, the corner is greater than 90°

Why? Since 32 + 42 = 52, then the triangle is a right triangle by the converse of the Pythagorean Theorem.

The Converse of the Pythagorean Theorem

5. The measures of three sides of a triangle are given below. Determine whether the

triangle is a right triangle. , 3, and 8

Which side is the longest?

The square root of 73 is about 8.5, therefore it must be the hypotenuse.

Plug your information into the Pythagorean Theorem. It doesn’t matter which number

is a or b.

9 + 64 = 7373 = 73

Since this is true, the triangle is a right triangle!! If it was not true, it

would not be a right triangle.

Sides: , 3, and 832 + 82 = ( ) 2

Three right triangles surround a shaded triangle; together they form a rectangle measuring 12 units by 14 units. The figure below shows some of the dimensions but is not drawn to scale.

Is the shaded triangle a right triangle? Provide proof for your answer.

12

14

7

5

59

12

14

7

5

59

No, the shaded triangle is not a right triangle.

The Distance FormulaThe distance formula is often memorized in the square root form with no connection to previous learning.

Many students do not make the connection that the distance formula

is simply the Pythagorean Theorem algebraically manipulated

by solving for d, which is the

hypotenuse of a right triangle..

Deriving the Distance Formula

C

The Distance from Point A to Point Bwould be equal to the length of the hypotenuseof triangle ABC.

C

Deriving the Distance Formula

Find the Distance Between:

1. Points A and B

2. Points B and C

3. Points A and C

Roland went on a hike to visit a cave in the mountains. To begin his hike he faced west and hiked for 3 miles. Then he turned to the south and traveled for 2 miles. After a water break Roland again continued west for 4 miles. Turning North he continued for 3 miles. Next Roland turned left for 2 miles, and then he took a right and continued on his hike for a final 6 miles until he discovered the location of the cave.

As “the crow flys”,

how far is the cave

from where Roland

started his hike?

West 3 milesSouth 2 milesWest 4 milesNorth 3 milesLeft 2 milesRight 6 miles

9

7

Pythagorean Theorem in the 3rd Dimension

2?

?

?

What is the longest curtain rod you can fit in this box?Note: Fishing poles only come in increments of tenth of a foot.

What is the longest curtain rod you can fit in this box?Note: Fishing poles only come in increments of tenth of a foot.

Pythagorean Theorem in the 3rd Dimension

2

?

The longest rod is 5.3 feet

Pythagorean Theorem in the 3rd Dimension

Derive a formula that will always work to find the diagonal of any rectangular prism.