NGSSS MA.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.

CCSS 8.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. The Common Core State Standards recognizes that mathematical topics cannot be taught in isolation. Each topic is interconnected with previous concepts and informs the understanding of later concepts. When students learn mathematics as a coherent whole with a structure that has connections between seemingly disparate topics, they become stronger mathematical thinkers. Rather than depend upon memorizing formulas and procedures that seem unconnected, students who understand the connections are empowered as mathematicians.

What is a right triangle? hypotenuse leg right angle leg 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.

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

Proof of the Pythagorean Theorem 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.

1. Find the length of the hypotenuse 122 + 162 = c2 144 + 256 = c2 400 = c2 Take the square root of both sides. 12 in 16 in Note: the square root of c2 is actually the absolute value of C, therefore C itself could be positive or negative. When applying the Pythagorean Theorem we only use the positive value because we are referring to the length of a triangle. However when students study further algebra they will need to know that the negative value may be a possible when solving quadratics, depending on the context of the problem. The hypotenuse is 20 inches long.

2. Find the length of the hypotenuse 52 + 72 = c2 25 + 49 = c2 74 = c2 Take the square root of both sides. 5 cm 7 cm Note: the square root of c2 is actually the absolute value of C, therefore C itself could be positive or negative. When applying the Pythagorean Theorem we only use the positive value because we are referring to the length of a triangle. However when students study further algebra they will need to know that the negative value may be a possible when solving quadratics, depending on the context of the problem. 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.

4. Find the length of the missing leg. 4 cm 10 cm 42 + b2 = 102 16 + b2 = 100 -16 -16 b2 = 84 The leg is about 9.2 cm long.

5. Find the length of the missing leg. a2 + 122 = 132 a2 + 144 = 169 -144 -144 a2 = 25 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. 24 ft. 4 ft. 6.9 ft. 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? http://www.baseball-almanac.com/stadium/baseball_field_construction.shtml

The Converse of the Pythagorean Theorem

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: 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.

Which side is the longest? 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.

Sides: , 3, and 8 32 + 82 = ( ) 2 9 + 64 = 73 73 = 73 Since this is true, the triangle is a right triangle!! If it was not true, it would not be a right triangle.

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. 12 14 7 5 9 Is the shaded triangle a right triangle? Provide proof for your answer.

No, the shaded triangle is not a right triangle. 12 14 7 5 9 No, the shaded triangle is not a right triangle.

The Distance Formula The 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 The Distance from Point A to Point B would be equal to the length of the hypotenuse of triangle ABC. C

Deriving the Distance Formula

Find the Distance Between: Points A and B Points B and C Points A and C

As “the crow flys”, how far is the cave from where Roland 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?

9 7 West 3 miles South 2 miles West 4 miles North 3 miles Left 2 miles Right 6 miles 9 7

Pythagorean Theorem in the 3rd Dimension What is the longest curtain rod you can fit in this box? Note: Fishing poles only come in increments of tenth of a foot. ? ? ? 2

Pythagorean Theorem in the 3rd Dimension What is the longest curtain rod you can fit in this box? Note: Fishing poles only come in increments of tenth of a foot. ? Square root of 29 is approx: 5.38516480713 which would round to 5.4, however in the context of this problem, if the pole was 5.4 feet, it would not fit in the box. 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.