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EXPLORE….. Using one of each color of the Cuisenaire Rods® try to build a right triangle.

1.Use the Cuisenaire Rods® to build a square on each side of the triangle on your paper.

2. With your group, find three different ways of showing that the combined area of the two smaller squares is the same as the largest square. (Rearrange the pink and green rectangles so that they fit ON TOP OF the yellow square.)

4 16 u u 2

4 16 u u u 2

Proof

Pythagoras is best known for the Pythagorean Theorem, which relates the side lengths of a right triangle.

The two sides that make up the right angle are called legs. The side opposite the right angle is the hypotenuse. leg hypotenuse right angle The Pythagorean Theorem a b c

In a right triangle, if a and b are the measures of the legs and c is the hypotenuse, then a 2 + b 2 = c 2. Note: The hypotenuse, c, is always the longest side.

Baseball Problem A baseball “diamond” is really a square. You can use the Pythagorean theorem to find distances around a baseball diamond.

Baseball Problem The distance between consecutive bases is 90 feet. How far does a catcher have to throw the ball from home plate to second base?

Baseball Problem To use the Pythagorean theorem to solve for x, find the right angle. Which side is the hypotenuse? Which sides are the legs? a 2 + b 2 = c 2 Now use: a 2 + b 2 = c 2

Baseball Problem Solution The hypotenuse is the distance from home to second, or side x in the picture. The legs are from home to first and from first to second. Solution: x 2 = = 16,200 x = ft

Check It Out: Example 1A 5 12 c 13 = c Pythagorean Theorem Substitute for a and b. a 2 + b 2 = c = c = c = c Simplify powers. Solve for c; c = c 2. Find the length of the hypotenuse to the nearest hundredth.

10 b 26 Finding c (HYPO) – square numbers, ADD, then take the square root! Finding a or b (LEG) – square numbers, SUBTRACT, then take the square root! STEP One: Identify the hypo and legs! COPY THIS INTO YOUR NOTES!

10 b 26 = 24 (a)(a) (c)(c)

1m 8m c b= a= c²=a²+ b² c²=1²+ 8² c²= c²=65 ? Using Pythagoras’ Theorem

Example 1 c 12cm 9cm a b a²+ b²= c 2 12²+ 9²= c² =c² c²= 225 c = √ 225= 15cm

c 6m 4m s a b a²+ b²= c 2 4²+ 6²= c² = c² c²= 52 c = √ 52 =7.2m (1 d.p.) Example 2

7m 5m h c a b a²+ b²= c² a²+ 5²= 7² a² + 25 = 49 ? Finding the shorter side

a² + 25= 49 We need to get a² on its own. Remember, change side, change sign! Finding the shorter side - 25 a²= = a²= 24 a = √ 24 = 4.9 m (1 d.p.)

169 = w² + 36 c w 6m 13m a b c²= a²+ b² 13²= a²+ 6² 169 – 36 = a² a = √ 133 = 11.5m (1 d.p.) a²= 133 Example = a² + 36 Change side, change sign!

c b c²= a²+ b² 11²= 9²+ b² 121 = 81 + b² 121 – 81 = b² b = √ 40 = 6.3cm (1 d.p.) b²= 40 a 9cm P 11cm R Q Example 2 81 Change side, change sign!

You can use The Pythagorean Theorem to solve many kinds of problems. Suppose you drive directly west for 48 miles, 48

Then turn south and drive for 36 miles

How far are you from where you started? ?

48 2 Using The Pythagorean Theorem, c =c2c2

Why? Can you see that we have a right triangle? c =c2c2

Which side is the hypotenuse? Which sides are the legs? c =c2c2

Then all we need to do is calculate:

And you end up 60 miles from where you started So, since c 2 is 3600, c is60.

1. The measures of three sides of a triangle are given below. Determine whether each triangle is a right triangle. 7, 3, and 8 Which side is the biggest? 8 This must be the hypotenuse (c). Plug your information into the Pythagorean Theorem. It doesn’t matter which number is a or b.

= 64 ? 58 = 64 ? This is NOT true, it is NOT a right triangle. Sides: 7, 3, and = 8 2

Determine whether the triangle is a right triangle given the sides 9, 15, and 12 1.Yes 2.No 3.Purple = = 225

Converse of the Pythagorean Theorem Theorem 8-2: Converse of the Pythagorean Theorem – If the square of the lengths of one side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle. If a 2 +b 2 =c 2, then the triangle is a right triangle.

If, complete each statement. Applying the Converse of the Pythagorean Theorem Are the following triangles right triangles? EXPLAIN = = = 7225 YES! = = = 784 NO!

Classifying Triangles Determine if the Triangle is a Right Triangle Given a triangle with sides 6, 11, and 14 how can you classify the triangle? Given a triangle with sides 7, 8, and 9 how can you classify the triangle? = = = 196 NO! = = = 81 NO!

Application The Parks Department rents paddle boats at docks near each entrance to the park. To the nearest meter how far is it to paddle from one dock to the other?

Use the Pythagorean Theorem to determine if the following are right triangles cm, 60 cm, 61 cm 2. 5 ft, 12 ft, 15 ft in, 9 in, 15 in cm, 20 cm, 48 cm