BELL-WORK TB pg 616 # 32,34,35

Return Work

HW 4.4(d) Due Monday: PW 10-1 #13-16, 20-21, 31, 33

HW 4.4(c) Solutions 13. 6 + 4√3 OR 4√3 + 6 7 – 2√7 OR -2√7 + 7 4 4 10√5 – 10√2 OR 10(√5 – √2) 4√37 – 4√5 OR 4(√37 – √5)

The nth Root One very important application of radicals is to the nth root. You are already familiar with the 2nd or square root, which means… what number squared gives the number under the radical. What do you think the 3rd or cube root means? what number cubed gives the number under the radical. (cube root on TI) What do you think the 4th root means? what number multiplied by itself 4 times gives the number under the radical. (xth on TI)

Guiding question: What is the Pythagorean Theorem?

Applications of Radicals Another very important application of radicals is the Pythagorean Theorem. Developed by the ancient mathematician Pythagoras, the Pythagorean Theorem is used to find missing measures of right triangles. Recall: What is a right triangle? A triangle with one 90o angle. What is the name given to the longest side of a right triangle? The hypotenuse is the longest side of a right triangle. The other two sides are known as legs.

Lengths of Right Triangles Which side of the triangle below is the hypotenuse?

Lengths of Right Triangles Note: the longest side of a triangle faces the largest angle. The right angle box points to the hypotenuse.

Lengths of Right Triangles Label the triangle below:

Lengths of Right Triangles Find the square of each length. What do you notice when you add the squares of the legs? Pythagoras discovered that this happens in every right triangle!

The Pythagorean Theorem Pythagoras discovered that the sum of the squares of the legs is equal to the square of the hypotenuse. Said mathematically: a2 + b2 = c2 ‘a’ and ‘b’ are the lengths of the legs of the right triangle and ‘c’ is the length of the hypotenuse. ‘c’ is the most important letter in the Pythagorean Theorem.

Pythagorean Theorem What is the length of the missing side of the triangle above? a2 + b2 = c2 82 + 152 = c2 64 + 225 = c2 289 = c2 c = 17

Pythagorean Theorem What is the length of the missing side of the triangle above? a2 + b2 = c2 32 + b2 = 52 9 + b2 = 25 b2 = 25 – 9 b2 = 16 b = 4

Real-World Pythagorean Theorem A 16-ft ladder is placed 4 ft from the base of a building. How high on the building will the ladder reach (to the nearest hundredth)? 42 + b2 = 162 16 + b2 = 256 b2 = 240 b = 15.49

Real World Pythagorean Theorem A pigeon leaves its nest in New York City and flies 5 km due east. The pigeon then flies 3 km due north. How far is the pigeon from its nest (to the nearest tenth)? 52 + 32 = c2 34 = c2 c = 5.8 km

Real World Pythagorean Theorem A ladder is 25 ft long. The ladder needs to reach to a window that is 24 ft above the ground. How far away from the building should the bottom of the ladder be placed? 242 + b2 = 252 b = 7 ft

Pythagorean Theorem Determine whether the lengths 5 in., 5 in., and 7 in. can be sides of a right triangle. 52 + 52 = 72 50 = 49 FALSE No the lengths cannot be the sides of a right triangle.

Pythagorean Theorem Determine whether the lengths 10 cm, 24 cm, and 26 cm can be sides of a right triangle. 102 + 242 = 262 676 = 676 TRUE Yes the lengths can be the sides of a right triangle.

Who wants to answer the Guiding question? What is the Pythagorean Theorem?