The Pythagorean Theorem and the Distance Formula Section 4.4

 Use the Pythagorean Theorem.  Use the Distance Formula.

Key Vocabulary Leg Hypotenuse Pythagorean Theorem Pythagorean Triple Distance Formula

Theorems 4.7 Pythagorean Theorem

Parts of a Right Triangle Longest side is the hypotenuse, side c (opposite the 90 o angle). The other two sides are the legs, sides a and b. Pythagoras developed a formula for finding the length of the sides of any right triangle.

HISTORY

Pythagoras Little is known about the life of Pythagoras. He was born about 569 BC on the Aegean island of Samos. Died about 475 BC. Studied in Egypt and Babylonia.

Pythagoras Found the Pythagorean Brotherhood in southern Italy during the sixth century B.C. Considered the first true mathematician. Used mathematics as a means to understand the natural world. Theorized that everything, physical and spiritual, had been assigned its alotted number and form. “Everything is number.”

Discoveries Discovered the existence of irrational numbers. The Pythagorean Theorem. He was the first to distinguish between prime and composite numbers. Discovered ideas in music and astronomy. d² = h² + a² + b ²

How Pythagoras Died Pythagoras died in a fire started by an angry mob. Only a few members of the Pythagorean Brotherhood survived.

Theorem The Pythagorean Theorem The Pythagorean Theorem In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. Example: (hypotenuse) 2 =(leg) 2 +(leg) 2

PYTHAGOREAN THEOREM INFORMAL PROOFS

Informal Proof #1 Inscribe a square within the square.

a b a a a b b b c c c c Informal Proof #1

a b a a a b b b c c c c a b c

b a a a b b c c c a b c

b a a a b b c c c a b c a b c

b a a b c c a b c a b c

b a a b c c a b c a b c b a c

a b c a b c a b c b a c

a b c a b c a b c b a c a b c

a b c a b c b a c a b c

a b c a b c b a c a b c

a b c a b c b a c a b c a b c

a b c b a c a b c a b c a b c

b a c a b c a b c a b c a b c

b a c a b c a b c a b c b a c

a b c a b c a b c b a c

a b a a a b b b c c c c Informal Proof #2 a + b Total White Yellow Area Area Area - = - =

Inform Proof #3

base height 1.We start with half the red square, which has Area = ½ base x height 2.We move one vertex while maintaining the base & height, so that the area remains the same. This is called a SHEAR. 3.We rotate this triangle, which does not change its area. base height 4.We mark the base and height for this triangle. (Area of green square)+ (Area of red square)= Area of the blue square The Pythagorean Theorem:

base height 5.We now do a shear on this triangle, keeping the same area. Remember that this pink triangle is half the red square. Half the red square. (Area of green square)+ (Area of red square)= Area of the blue square The Pythagorean Theorem: 1.We start with half the red square, which has Area = ½ base x height 2.We move one vertex while maintaining the base & height, so that the area remains the same. This is called a SHEAR. 3.We rotate this triangle, which does not change its area. 4.We mark the base and height for this triangle.

(Area of green square)+ (Area of red square)= Area of the blue square The Pythagorean Theorem: 6.The other half of the red square has the same area as this pink triangle, so if we copy and rotate it, we get this. So, together these two pink triangles have the same area as the red square. 7.We now take half of the green square and transform it the same way. Half the red square. We end up with this triangle, which is half of the green square. Half the green square. 9.Together, they have they same area as the green square. So, we have shown that the red & green squares together have the same area as the blue square. Shear Rotate Shear 8.The other half of the green square would give us this.

(Area of green square)+ (Area of red square)= Area of the blue square The Pythagorean Theorem: 6.The other half of the red square has the same area as this pink triangle, so if we copy and rotate it, we get this. So, together these two pink triangles have the same area as the red square. 7.We now take half of the green square and transform it the same way. Half the red square. We end up with this triangle, which is half of the green square. Half the green square. 9.Together, they have they same area as the green square. So, we have shown that the red & green squares together have the same area as the blue square. We’ve PROVEN the Pythagorean Theorem! Shear Rotate Shear 8.The other half of the green square would give us this. WWWW eeee ’’’’ vvvv eeee P P P P rrrr oooo vvvv eeee nnnn tttt hhhh eeee PPPP yyyy tttt hhhh aaaa gggg oooo rrrr eeee aaaa nnnn TTTT hhhh eeee oooo rrrr eeee mmmm

Find the length of the hypotenuse. ANSWER The length of the hypotenuse is 13. SOLUTION (hypotenuse) 2 = (leg) 2 + (leg) 2 Pythagorean Theorem c 2 = Substitute. c 2 = Multiply. c 2 = 169 Add. c = 13 Solve for c. Find the positive square root. c 2 = 169 Example 1

Find the unknown side length. ANSWER The side length is about SOLUTION (hypotenuse) 2 = (leg) 2 + (leg) 2 Pythagorean Theorem 14 2 = b 2 Substitute. 196 = 49 + b 2 Multiply. 147 = b 2 Simplify ≈ b Approximate with a calculator. 196 – 49 = 49 + b 2 – 49 Subtract 49 from each side. Find the positive square root. 147 = b2b2 Example 2

ANSWER 8 8 about 10.6 Find the unknown side length Your Turn:

Example 3a A. Find x. The side opposite the right angle is the hypotenuse, so c = x. a 2 + b 2 = c 2 Pythagorean Theorem = c 2 a = 4 and b = 7

Example 3a 65= c 2 Simplify. Take the positive square root of each side. Answer:

Example 3b B. Find x. The hypotenuse is 12, so c = 12. a 2 + b 2 = c 2 Pythagorean Theorem x = 12 2 b = 8 and c = 12

Example 3b Take the positive square root of each side and simplify. x = 144Simplify. x 2 = 80Subtract 64 from each side. Answer:

Your Turn: A. Find x. A. B. C. D.

Your Turn: B. Find x. A. B. C. D.

More Examples: 1) A=8, C =10, Find B 2) A=15, C=17, Find B 3) B =10, C=26, Find A 4) A=15, B=20, Find C 5) A =12, C=16, Find B 6) B =5, C=10, Find A 7) A =6, B =8, Find C 8) A=11, C=21, Find B A B C B = 6 B = 8 A = 24 C = 25 B = 10.6 A = 8.7 C = 10 B = 17.9

Your Turn: How high up on the wall will a twenty-foot ladder reach if the foot of the ladder is placed five feet from the wall? Answer: 19.4

Pythagorean Triples Pythagorean Triples Three whole numbers that work in the Pythagorean formulas are called Pythagorean Triples. The largest number in each triple is the length of the hypotenuse. Pythagorean triples are not the only possible side lengths for a right triangle. They give the triangles where all the lengths are whole numbers, but the side lengths could be any real numbers.

Pythagorean Multiples If you multiply the lengths of all three sides of any right triangle by the same number, then the resulting triangle is a right triangle. In other words, if a 2 + b 2 = c 2, then (an) 2 + (bn) 2 = (cn) 2. Therefore, additional pythagorean triples can be found by multiplying each number in a known triple by the same factor.

Pythagorean Triples Multiples

Primitive Pythagorean Triples primitive Pythagorean triple A set of Pythagorean triples is considered a primitive Pythagorean triple if the numbers are relatively prime; that is, if they have no common factors other than 1. You need know the first 4 primitives: 3-4-5, , ,

Example 4 Use a Pythagorean triple to find x. Explain your reasoning.

Example 4 Notice that 24 and 26 are multiples of 2 : 24 = 2 ● 12 and 26 = 2 ● 13. Since 5, 12, 13 is a Pythagorean triple, the missing leg length x is 2 ● 5 or 10. Answer:x = 10 Check: = 26 2 Pythagorean Theorem ? 676 = 676Simplify.

Your Turn: A.10 B.15 C.18 D.24 Use a Pythagorean triple to find x.

More Practice Use Pythagorean Triples to find each missing side length. Primitive: X=26 Primitive: X=50 Primitive: X=15

THE DISTANCE FORMULA

Let's find the distance between two points. So the distance from (-6,4) to (1,4) is 7. If the points are located horizontally from each other, the y coordinates will be the same. You can look to see how far apart the x coordinates are. (1,4)(-6,4) 7 units apart distance equals |x 2 -x 1 |=|1-(-6)|=7

What coordinate will be the same if the points are located vertically from each other? So the distance from (-6,4) to (-6,-3) is 7. If the points are located vertically from each other, the x coordinates will be the same. You can look to see how far apart the y coordinates are. (-6,-3)(-6,4) 7 units apart distance equals |y 1 -y 2 |=|-3-4|=7

But what are we going to do if the points are not located either horizontally or vertically to find the distance between them? Let's add some lines and make a right triangle. This triangle measures 4 units by 3 units on the sides. If we find the hypotenuse, we'll have the distance from (0,0) to (4,3) Let's start by finding the distance from (0,0) to (4,3) ? 4 3 The Pythagorean Theorem will help us find the hypotenuse 5 So the distance between (0,0) and (4,3) is 5 units.

Now let's generalize this method to come up with a formula so we don't have to make a graph and triangle every time. Let's add some lines and make a right triangle. Solving for c gives us: Let's start by finding the distance from (x 1,y 1 ) to (x2,y2)(x2,y2) ? x 2 - x 1 y 2 – y 1 Again the Pythagorean Theorem will help us find the hypotenuse (x 2,y 2 ) (x1,y1)(x1,y1) This is called the distance formula

Let's use it to find the distance between (3, -5) and (-1,4) (x1,y1)(x1,y1)(x2,y2)(x2,y2) CAUTION! You must do the brackets first then powers (square the numbers) and then add together BEFORE you can square root Don't forget the order of operations! means approximately equal to found with a calculator Plug these values in the distance formula

(hypotenuse) 2 = (leg) 2 + (leg) 2 (AB) 2 = Substitute. (AB) 2 = Multiply. (AB) 2 = 25 Add. AB = 5 Simplify. Find the distance between the points A(1, 2) and B(4, 6). SOLUTION Draw a right triangle with hypotenuse AB. BC = 6 – 2 = 4 and CA = 4 – 1 = 3. Use the Pythagorean Theorem. Find the positive square root. (AB) 2 =25 Example 5

Find the distance between D(1, 2) and E(3, 2). SOLUTION Begin by plotting the points in a coordinate plane. x 1 = 1, y 1 = 2, x 2 = 3, and y 2 = –2. ≈4.5 Approximate with a calculator. ANSWER The distance between D and E is about 4.5 units. The Distance Formula DE = (x 2 – x 1 ) 2 + (y 2 – y 1 ) 2 Substitute. = (3 – 1) 2 + (–2 – 2) 2 Simplify. = (–4) 2 Multiply. = Add. = 20 Example 6

ANSWER 5 about 6.3 ANSWER about 5.1 Find the distance between the points Your Turn:

Find the Distance between Points A and B. Distance Formula Example 7

Substitute Simplify. Square the numbers. Simplify Your Turn:

Given: Q (3, 8) and R (-4, 6) Find: the length of segment QR. Formula: Substitute: Simplify Parentheses: Square Numbers: Solution:

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Assignment Sec. 4.4, Pg 195 – 198: 1 – 31 odd, 35 – 39 odd, Quiz 2 1 – 7 all