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Warm Up for TrigA- An intro to Trig

Warm Up for TrigA- An intro to Trig. A 36m tree cracked at the hinge. The tip of the tree hit the ground 24m from the base. How many meters up from the base of the tree is the crack? Are the following triples, yes or no? A. 8,15,17 B. 10, 20, 24

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Warm Up for TrigA- An intro to Trig

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  1. Warm Up for TrigA- An intro to Trig A 36m tree cracked at the hinge. The tip of the tree hit the ground 24m from the base. How many meters up from the base of the tree is the crack? Are the following triples, yes or no? A. 8,15,17 B. 10, 20, 24 How high on a building will a 15-foot ladder touch if the foot of the ladder is 5 feet from the building? Find the other leg: one leg is 5 cm and the hypotenuse is 13 cm.

  2. Answers • x2 + (24)2 = (36-x)2 = 10 2A. Yes 2B. No • 14.1 ft • 12 cm

  3. Special Right Triangles There are two special right triangles! We will use the Pythagorean Theorem to discover the relationships between the sides of the two special triangles.

  4. The first special triangle is an isosceles triangle, also called a • Each isosceles triangle is half a square so they appear in math and engineering frequently. An isosceles right triangle

  5. Investigation 1 • Step 1 Sketch an isosceles right triangle. Label the legs s and the hypotenuse h • Step 2 Pick any integer for s, the length of the legs. Use the Pythagorean Theorem to find h. • Step 3 Repeat Step 2 with a different integer for s. What do you notice? Do you see any pattern in the relationship between s and h?

  6. Isosceles Right Triangle Conjecture or45-45-90 Rule In an isosceles right triangle, if the legs have length s, then the hypotenuse has length _______ Think: side – side – side

  7. Lets try a few: Find the missing sides 1) 5 8 7 5

  8. 30 60 The other special triangle is a If you fold an equilateral triangle along one of its altitudes you get a 30-60-90 triangle. Therefore, a 30-60-90 triangle is one half an equilateral triangle so it appears in math and engineering frequently as well. Let’s see if there is a way to figure out the lengths of the legs.

  9. C A B D Let’s use some deductive reasoning to figure this one out. • Triangle ABC is equilateral, and CD is an altitude Step 1 What are and ? What are the and ? What are the and ?

  10. C A B D More steps to help us understand… • Step 2 Is ? Why? • Yes because of SAS, ASA, or SAA • Step 3 Is ? Why? • How do AC and AD compare? • Yes, CPCTC, AC = 2AD • In a 30-60-90 triangle, will this relationship between the hypotenuse and the shorter leg always hold true? Explain • Yes, all 30-60-90 triangles are similar.

  11. 60 30 x So, now we can use the Pythagorean Theorem to find the unknown side. A couple more… • Step 4 Sketch a 30-60-90 triangle. Choose an integer for the short side…let’s use 5. What do we know about the hypotenuse? 10 5 It is two times bigger than the short side, so the hypotenuse is…

  12. 60 30 10 5 Try one on your own! Do you see a pattern between the two legs?

  13. 60 30 30-60-90 Triangle Conjecture In a 30-60-90 triangle, if the shorter side has length s, then the longer leg has length _______ and the hypotenuse has length _____ Think: side – side – 2 · side

  14. Let’s practice someUse your own paper Find the unknown length 1. 3. y x 10 x 45 30 12 2. 4. y x 60 18 x

  15. Summary Homework: pg 477# 1-6

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