Dr. Fowler – CCM1A Unit 1 – Lesson 9 Pythagorean Theorem

Warmup Solve for x x2+7=43 Ans: x = ±6 64+x2=164 Evaluate for a = 12, b = 5, c = 13 a2 + b2 c2 – b2 Ans: x = ±6 Ans: x = ±10 Ans: 169 Ans: 144

550 B.C.

Pythagoras Pythagoras was a Greek philosopher and religious leader. He was responsible for many important developments in maths, astronomy, and music.

The Secret Brotherhood His students formed a secret society called the Pythagoreans. As well as studying math, they were a political and religious organization. Members could be identified by a five pointed star they wore on their clothes.

The Secret Brotherhood They had to follow some unusual rules. They were not allowed to wear wool, drink wine or pick up anything they had dropped! Eating beans was also strictly forbidden!

Here we have a triangle with the lengths of each of the three sides 5 4 3

Let’s take the lengths of each side and make a square for each of them 5 4 3

Let’s find the area of each square? 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 2 3 4 16 17 18 19 20 5 6 7 8 21 22 23 24 25 9 10 11 12 13 14 15 16 1 2 3 4 5 6 7 8 9

Now, let’s add the two smaller areas together. 25 16 + 9

It turns out this is true for every right triangle Notice how the sum of the two smaller squares equals the larger square? 25 It turns out this is true for every right triangle + = ? 9 16

The Pythagorean Theorem states: “The sum of the squares of the legs of a right triangle are equal to the square of the hypotenuse.” + = 25 9 16

The Pythagorean Theorem a2 + b2 = c2 c a b Where a and b are legs and c is the hypotenuse The Hypotenuse is always the BIGGEST value

Pythagorean Theorem hypotenuse leg leg The hypotenuse is always the longest side of a right triangle and is always opposite the right angle.

Pythagorean Theorem What is the value of the missing side? 5 12

Pythagorean Theorem What is the value of the missing side? 15 9

Using a2 + b2 = c2 a2 + b2 = c2 152 + 202 = c2 225 + 400 = c2 625 = c2 Looking for length of the hypotenuse a2 + b2 = c2 152 + 202 = c2 225 + 400 = c2 625 = c2 x 15 20

Using a2 + b2 = c2 a2 + b2 = c2 62 + b2 = 102 36 + b2 = 100 -36 -36 10 Find the Leg a2 + b2 = c2 62 + b2 = 102 36 + b2 = 100 -36 -36 b2 = 64 b = 8 10 6 x

Example 1. Find the length of AC. Hypotenuse 16 B C 12 Solution : AC2 = 122 + 162 (Pythagoras’ Theorem) AC2 = 144 + 256 AC2 = 400 AC = 20

d Example 2. Find the length of diagonal d . Solution: 10 24 d Solution: d2 = 102 + 242 (Pythagoras’ Theorem)

Applying the Pythagorean Theorem

Application of Pythagoras’ Theorem 1.A car travels 16 km from east to west. Then it turns left and travels a further 12 km. Find the displacement between the starting point and the destination point of the car. 16km N 12km ?

Worksheet?

What is Volume? Volume is the measure of the capacity of a container. It is the measure of how much a container of a particular shape will hold - liquids, dry substances, etc.

AC2 = AB2 + BC2 (Pythagoras’ Theorem) AC2 = 162 + 122 AC2 = 400 16 km 12 km A B C Solution : In the figure, AB = 16 BC = 12 AC2 = AB2 + BC2 (Pythagoras’ Theorem) AC2 = 162 + 122 AC2 = 400 AC = 20 The displacement between the starting point and the destination point of the car is 20 km

2. The height of a tree is 5 m. The distance between the top of it and the tip of its shadow is 13 m. Find the length of the shadow L. Solution: 132 = 52 + L2 (Pythagoras’ Theorem) L2 = 132 - 52 L2 = 144 L = 12 5 m 13 m L

A 15 foot ladder leans up against a building A 15 foot ladder leans up against a building. The foot of the ladder is 5 feet from the base of the building. How high up the wall, to the nearest foot does the ladder reach? Draw a picture: 15 x 5

Solving the problem The ladder reaches 14 feet up the wall. a2 + b2 = c2 Write formula x2 + 52 = 152 Substitute in x2 + 25 = 225 Solve for x - 25 -25 x2 = 200 x = 14.142135 Check: How am I to leave my answer? The ladder reaches 14 feet up the wall.

Pythagoras Theorem H B L A boat sails due East from a Harbour (H), to a marker buoy (B),15 miles away. At B the boat turns due South and sails for 6.4 miles to a Lighthouse (L). It then returns to harbour. What is the total distance travelled by the boat? H B L 15 miles 6.4 miles Total distance travelled = 21.4 + 16.4 = 37.7 miles

Pythagoras Theorem A 12 ft ladder rests against the side of a house. The top of the ladder is 9.5 ft from the floor. How far is the base of the ladder from the house? 12 ft 9.5 ft L

You try this one in your notes. TAKE NOTES You try this one in your notes. x Find x 5 ft 12 ft 52 + 122 = x2 25 + 144 = x2 169 = x2 Answer: x = 13

Try this one in your notes… x 15 20 Solve for x. Round your answer to the nearest hundredth if necessary. Answer: 25

Try this one in your notes… 7 12 x Solve for x. Round your answer to the nearest hundredth if necessary. Answer: 13.89

Try this one in your notes… 5 x 3 Solve for x. Round your answer to the nearest hundredth if necessary. Answer: 4