Warm Up Classify each triangle by its angle measures. 1. 2. 3. Simplify 4. If a = 6, b = 7, and c = 12, find a2 + b2 and find c2. Which value is greater? acute right 12 85; 144; c2

Objectives Use the Pythagorean Theorem and its converse to solve problems. Use Pythagorean inequalities to classify triangles. Justify and apply properties of 45°-45°-90° triangles. Justify and apply properties of 30°- 60°- 90° triangles.

pythagorean theorem

Chou-pei Suan-king Chinese Book from 1200-600 B.C. Many years after Chinese in 560 B.C., Pythagoras made a formal proof…the Pythagorean Theorem that's over 1,445 years ago!

Always across from the right angle. Pythagorean Theorem HYPOTENUSE Always across from the right angle. LEG LEG

Pythagorean Theorem c a b The square of the hypotenuse is equal to the sum of the square of the other two sides. ONLY FOR RIGHT TRIANGLES

http://www.usna.edu/MathDept/mdm/pyth.html

16 ft 12 ft

m LEG HYP LEG 14 m

Round to the nearest tenth 3m 4 m

A set of three nonzero whole numbers a, b, and c such that a2 + b2 = c2 is called a Pythagorean triple.

x and y 1 and 2 1 and 3 3 and 4 You can create Pythagorean Triples. Choose 2 integers, x and y. x and y 1 and 2 1 and 3 3 and 4 Create your own

If c is the measure of the hypotenuse, find each missing measure If c is the measure of the hypotenuse, find each missing measure. Round to the nearest tenth, if necessary. 1. 2.

Converse of the Pythag Thrm If , then the triangle is a right triangle. Converse: the hypothesis & conclusion are interchanged Original Pythag: If you have a right triangle, then

The measures of 3 sides for a triangle are given The measures of 3 sides for a triangle are given. Determine whether each triangle is a right triangle. 1. 20, 21, 28 2. 10, 24, 26 Check for a2 + b2 = c2 The legs are always the 2 smaller sides. no yes

Play Ball! 2nd Base 90 ft 90 ft 90 ft 90 ft Home Plate How far does a catcher have to throw when he throws the ball from home plate to second base?

Special Right Triangles 45-45-90 And 30-60-90

An isosceles right triangle Each isosceles triangle is half a square, so they show up a lot in math and engineering.

Pick any integer for l. Use Pythagorean Theorem for find h. Let’s look for a shortcut for finding the length of an unknown side in a 45-45-90 triangle: Pick any integer for l. Use Pythagorean Theorem for find h. h l l

This is our reference triangle for the 45-45-90. In an isosceles right triangle, if the legs have length l, then the hypotenuse has length ____.   1 1 This is our reference triangle for the 45-45-90.

EX: 1 Solve for x x 3 3

EX: 2 Solve for x x 5 5

EX: 3 Solve for x x 3 45

EX: 3 Solve for x x 45

EX: 3 Solve for x 45 x

30-60-90

If you fold an equilateral triangle along one of its altitudes, the triangles you get are 30-60-90.

This is our reference triangle for the 30-60-90 triangle. 2 1 30 This is our reference triangle for the 30-60-90 triangle.

Ex: 1 60 x 8 30 y

Ex: 2 Solve for x 60 30 24 x x = 12

Ex: 3 30 14 y 60 x

Ex: 3 30 x y 60 20

Ex: 4 x 60 30 y y = 10 x = 5

Objectives Use the Pythagorean Theorem and its converse to solve problems. Use Pythagorean inequalities to classify triangles. Justify and apply properties of 45°-45°-90° triangles. Justify and apply properties of 30°- 60°- 90° triangles.

Always across from the right angle. Pythagorean Theorem _______________ Always across from the right angle. ____ ____

16 ft 12 ft

m LEG HYP LEG 14 m

Round to the nearest tenth 3m 4 m

A set of three nonzero whole numbers a, b, and c such that a2 + b2 = c2 is called a ____________ _______.

If c is the measure of the hypotenuse, find each missing measure If c is the measure of the hypotenuse, find each missing measure. Round to the nearest tenth, if necessary. 1. 2.

Converse of the Pythag Thrm If , then the triangle is a _______________ triangle. Converse: the hypothesis & conclusion are interchanged Original Pythag: If you have a right triangle, then

The measures of 3 sides for a triangle are given The measures of 3 sides for a triangle are given. Determine whether each triangle is a right triangle. 1. 20, 21, 28 2. 10, 24, 26

Play Ball! 2nd Base 90 ft 90 ft 90 ft 90 ft Home Plate How far does a catcher have to throw when he throws the ball from home plate to second base?

Special Right Triangles 45-45-90 And 30-60-90

This is our reference triangle for the 45-45-90. In an isosceles right triangle, if the legs have length l, then the hypotenuse has length ____.   1 1 This is our reference triangle for the 45-45-90.

EX: 1 Solve for x x 3 3

EX: 2 Solve for x x 5 5

EX: 3 Solve for x 45 3 x

EX: 3 Solve for x 45   x

EX: 3 Solve for x 45 x

This is our reference triangle for the 30-60-90 triangle. 2 1 30 This is our reference triangle for the 30-60-90 triangle.

Ex: 1 60 8 x 30 y

Solve for x Ex: 2 30 x 24 60

Ex: 3 30 14 y 60 x

Ex: 3 30 15 y 60 x

Ex: 4 x 60 30 y

Ex: 4 x 60 30 y