Triangle a b c a² b² c²

Blue* 5 12 13 25 144 169

Green 7 8 13 49 64 169

Orange* 3 4 5 9 16 25

Pink 6 5 6 36 25 36

Purple* 6 8 10 36 64 100

White* 7 24 25 49 576 625

Yellow* 8 15 17 64 225 289

Pythagoras and his Theorem

• Right Triangle: a triangle with exactly one right angle.

• Legs: the sides of a triangle that form the right angle.

• Hypotenuse: the longest side, its located across from the right angle.

c

a

b

The legs are labeled a & b and the

hypotenuse is ALWAYS labeled c.

Pythagoras discovered that the sum of the squares of the two

legs in a right triangle is equal to the square of the hypotenuse.

That means, in any right triangle,

a² + b² = c² leg² + leg² =

hypotenuse²

The term The term “squared” “squared” comes from the comes from the area of a area of a square.square.

EX:EX:

3 “squared” 3 “squared” means 3x3 or means 3x3 or 9.9.

The area of a The area of a 3x33x3 square is 9 square is 9

Could a right triangle have sides that measure 3 cm, 4 cm, and 5

cm?a² + b² = c²3² + 4² ? 5² 9 + 16 ? 25 25 = 25

Yes, this is a right triangle because the Pythagorean Theorem works!

How about sides of 5, 6, and 7?

a² + b² = c²5² + 6² ? 7² 25 + 36 ? 49

61 ≠ 49

NO, this is a NOT right triangle because the Pythagorean Theorem

doesn’t work!

Is 15, 17, 8 a right triangle? Why or why not? Show Work!

a² + b² = c²8² + 15² ? 17² 64 + 225 ? 289

289 = 289

Yes, this is a right triangle because the Pythagorean Theorem works!

Using the Pythagorean Theorem to Find a Missing Side

Note: the missing side is the hypotenuse

a² + b² = c²

5² + 12² = c²

25 + 144 = c²

169 = c²

√169 = √c²

13 = c

What if you know the hypotenuse? You can use the theorem to find one of the legs.

a² + b² = c²9² + b² = 15² 81 + b² = 225-81 -81

b² = 144 √b² = √144 b = 12

When your answers don’t work out evenly, round to the

nearest TENTHTENTH . a² + b² = c² a² + 4² = 11² a² + 16 = 121

-16 -16 b² = 105

√b² = √105 b = 10.246

10.2m

Finally, we can use the Pythagorean Theorem to solve

real life word problems.

Jen hiked 8 miles east, then turned and hiked 6 miles south. How far was she from her starting

point?

DRAW A PICTURE!

Jen hiked 8 miles east, then turned and hiked 6 miles south. How far was she

from her starting point?

8 miles east

6 miles

south

?

a² + b² = c² 8² + 6² = c² 64 + 36 = c² 100 = c² √100 = √c² 10 = c

Jen was 10 miles from where she started.

Polygons

• Polygon: a closed figure formed by 3 or more line segments that intersect only at their verticies.

• Polygons are classified by the number of sides and angles they have

Polygons

3 sides:3 sides:

triangletriangle4 sides:4 sides:

quadrilateral quadrilateral 6 sides:6 sides:

hexagonhexagon5 sides:5 sides:

pentagonpentagon

7 sides:7 sides:

heptagonheptagon10 sides:10 sides:

decagondecagon9 sides:9 sides:

nonagonnonagon 8 sides:8 sides:

octagonoctagon

Regular Polygons • Regular Polygon: a polygon in which all the

sides are the same length and all the angles are the same measure.

Example:

Interior Angles

Sum of Interior Angle Formula:Sum of Interior Angle Formula:

(n – 2) * 180(n – 2) * 180

What if we wanted to know the measure of EACH interior angle of a regular pentagon?

How would we go about doing this?Discuss with your partner.

Find the measure of each interior angle of a pentagon.

108º108º

Find the measure of each interior angle of a hexagon.

120º120º

How many sides does a polygon have if the sum of its

interior angles is 1440º.

10 sides10 sides

Find the measure of the missing angle in the figure below.

93º93º