Section 9.4Geometry’s Most Elegant

Theorem“Pythagorean Theorem”

Pages 384-391

Rebecca Sproul

Pythagorean Theorem• As the plays of Shakespeare are to

literature, as the Mets are to Mr. Pricci, so is the Pythagorean Theorem to geometry. The Pythagorean is the rule for solving right triangles, it’s widely applied because every polygon can be divided into right triangles by diagonals and altitudes, and it enables many ideas (and objects) to fit together very simply. “Indeed it is elegant in concept and very powerful” (Rhoad 384).

• Theorem 69- The square of the measure of the hypotenuse of a right triangle is equal to the sum of the squares of the measures of the legs (Pythagorean Theorem)

• [ a2 + b2 = c2]• Sample:

• Use the Pythagorean Theorem

92 +122 = x2

81 + 144 = x2

225 = x2

225 = x

+\- 15 (reject -15)

x = 15

Another Look at the TheoremThe Pythagorean Theorem must work in any 90 degree triangle. This means that if you know two of the sides, you can always find the third one.

In the right triangle at the left, we know that: h2 = 62 + 82

Simplifying the squares gives: h2 = 36 + 64and then: h2 = 100     h = 10        (by doing the square root of 100)

Another Theorem

• Theorem 70: If the square of the measure of one side of a triangle is equal to the sum of the squares of the measures of the other two sides, then the angle opposite the longer side is a right angle.

If a2 + b2 = c2

Then ABC is a right triangle

and C is the right angle

Extension of Theorem 70

If in the diagram on the last slide we increased c while keeping

a and b the same, c would become larger. Thus, a

valuable extension of Theorem 70 can be stated:

• a2 + b2 > c2, then the is acute

• a2 + b2 = c2, then the is right• a2 + b2 < c2, then the is

obtuse

Extension of Theorem 70

• Examples: Given the following sides in a triangle, classify the triangle as acute, right, or obtuse

1. 9, 12, 15

92 + 122 _ 152

81 + 144 _ 225

225 = 225

Right

2. 11, 13, 7

72 + 112 _ 132

49 + 121 _ 169

170 > 169

Acute

3. 4 3, 6 3, 5 2

(4 3)2 + (5 2)2 _ (6 3)2

48 + 50 _ 108

98 < 108

Obtuse

Example Problem

• 1. Triangle ABC is equilateral with a perimeter of 18. Find x.

Since ABC is equilateral, divide its perimeter into 3 equal sides

Segment BD is the perpendicular bisector of segment AC, so AC can be divided by 2

BDC is a right angle (perpendicular lines form right angles), so we can use the Pythagorean Theorem to find x.

x2 + 32 = 62

x2 + 9 = 36

x2 = 27

x = 3 3

Example Problem• Draw both diagonals in the rhombus, and

since diagonal bisects each other in a rhombus, cut each diagonal in half.

• Diagonals in a rhombus are perpendicular, so the angles formed in the middle of the diagonals are right angles (perpendicular lines form right angles).

• Since we have right angles, use the Pythagorean Theorem to solve for x, one side of the rhombus. Once you find x, multiply it by 4 because all 4 sides in a rhombus are =.

2. RSTU is a rhombus with diagonals of 12 and 24. Find the perimeter of RSTU.

62 + 122 = x2

36 + 144 = x2

169 = x2

x = 13 (reject -13)

4x = 52 = perimeter of rhombus RSTU

Example Problem3. Domo (1) traveled 5 km north, 2 km east, 10 km north, and 6 km east to find another Domo (2). How far is Domo (1) from where he started?

Draw Domo (1)’s path to Domo (2) and label the distance traveled

Notice how the distance apart (x) forms a right triangle when the sides are drawn out

Label each side of the right triangle by adding the distance traveled of the corresponding sides of Domo (1)’s path

5 + 10 = 15

2 + 6 = 8

Now use the Pythagorean Theorem

82 + 152 = x2

64 + 225 = x2

289 = x2

x = 17 (reject -17) = distance apart from Domos

Example Problem

• 4. A square has a perimeter of 36. Find the length of its diagonal.

Since this is a square, all sides are =, so divide its perimeter (36) by how many sides it has (4) to get a single side of the square (9).

Squares contain right angles, so we can use the right triangle formed by the diagonal drawn in the square by doing the Pythagorean Theorem

92 + 92 = x2

81 + 81 = x2

162 = x2

x = 9 2 (reject the negative) = the diagonal

Example Problem• 5. Find x in the trapezoid

Draw new altitudes in the trapezoid to form a right triangle. The altitudes will have the same length (x)

Each altitude forms a right angle

Since the top of the trapezoid is parallel to the bottom, subtract the top from the bottom (27 – 17) to find the difference (10)

Then divide the difference by 2 to get each of the small legs of each right triangle (5)

Now use Pythagorean Theorem

x2 + 52 = 122

x2 + 25 = 144

x2 = 119

x = 119 (reject the negative) = the altitude

Example Problem6. How far up a wall will an 11m ladder reach, if the foot of the ladder must be 4m from the base of the wall?

Since the ladder against the wall forms a right triangle, the Pythagorean Theorem can be used

42 + x2 = 112

16 + x2 = 121

x2 = 105

x = 105

Sample Problem

1. Find x

Click for answer

46

Sample Problem

2. What type of triangle is demonstrated by the 3 sides in each set: obtuse,

acute, or right?

16, 24, 32

17, 9, 15

3, 5, 4

14, 50, 48

Click for answer

obtuse, acute,

acute, right

Sample Problem3.Domo walks 15 km north, 4 km west, 3 km north, 9 km west, and 2 km north and got very lost. How far is domo from where is first started?

Click for answer

5 17

Sample Problem

4. The lengths of the diagonals of a rhombus are in the ratio of 2:1. If the perimeter of the rhombus is 60, find the sum of the lengths of the diagonals.

Click for answer

18 5

Sample Problem5. Find the length of the altitude of an equilateral triangle with a side of 8.

4 3

Click for answer

Sample Problem                       6. What is the diagonal length of a TV screen whose dimensions are80 x 60 cm?

Click for answer

100

Sample Problem7. Find the altitude (length of a segment perpendicular to both bases) of the isosceles trapezoid shown.

8

Click for answer

Sample Problem8. Domo’s pet bird pecked at a 13-m wooden pole until is cracked and the upper part fell, with the top hitting the ground 7 m from the foot of the pole. Since the upper part had not completely broken off, Domo’s bird pecked away where the pole had cracked. How far was Domo’s bird above the ground?

Click for answer

2 15

Works Cited Page

Willis, Bill. "The Pythagorean Theorem." 1999 28 May 2008 <http://www.worsleyschool.net/science/files/pythagoras/pythagoreanth eorem2.html>.

Rhoad, Richard, and Milauskas, and Whipple. Geometry for Enjoyment and Challenge. New ed. Boston: McDougal Littell, 1991.