BELLWORK 1. Write a similarity statement comparing the two triangles. Simplify each radical. 2. 3. Solve each equation. 4. 5. 2x2 = 50

§ 8.1, Right Triangle Similarity Learning Targets I will use geometric mean to find segment lengths in right triangles. I will apply similarity relationships in right triangles to solve problems. Vocabulary geometric mean 8-1

Draw and write a similarity statement comparing the three triangles. The altitude of a right triangle will divide the triangle into two other triangles. All three triangles are similar. Draw and write a similarity statement comparing the three triangles. ∆UVW ~ ∆UWZ ~ ∆WVZ Z W

Geometric mean… For any two positive numbers, a and b, the geometric mean is the positive number x such that: In addition Find the geometric mean of 4 and 9 Find the geometric mean of 3 and 12. Write a proportion. = 3 x 12 x2 = 36 Cross-Product Property Find the geometric mean of 10 and 30 x = 36 x = 6

Corollary 8-1-2 The length of the altitude to the hypotenuse is the geometric mean of the lengths of the resulting segments . A B C D

Corollary 8-1-3 The altitude to the hypotenuse separates the hypotenuse so that the length of each leg of the right triangle is the geometric mean of the length of the adjacent hypotenuse segment and the length of the hypotenuse. B C D A

Find the values of x, y, and z. Can’t solve these two… But we can solve this one…

To estimate the height of a Douglas fir, Jim positions himself so that his lines of sight to the top and bottom of the tree form a 90º angle. His eyes are about 1.6 m above the ground, and he is standing 7.8 m from the tree. What is the height of the tree to the nearest meter?

SOHCAHTOA Some Old Horse Caught Another Horse Taking Oats Away

§8.2, Trigonometric Ratios Learning Targets I will find the sine, cosine, and tangent of an acute angle. I will use trigonometric ratios to find side lengths in right triangles and to solve real-world problems. Vocabulary trigonometric ratio sine cosine tangent

In trigonometry we will be using right triangles. There are two angles that are NOT the 90 angle and most times we will be referencing those two angles. Each angle will have its own opposite side, adjacent side and hypotenuse. A 8m 6m 10m B C

This ratio is called the tangent. B C Given A, The ratio of the leg opposite to A to the leg adjacent to A is fixed. This ratio is called the tangent. Tangent A = Length of leg opposite A Length of leg adjacent A Tan A = opposite adjacent

Each triangle is a different size but, the ratio of the opposite side to the adjacent side remains constant. So the tangent of 30° is the same regardless how big the triangle sides are. x 6 3 x 4 2 x 2 1

Write the tangent ratios for A and B. tan A 20 21 = opposite adjacent BC AC tan B 21 20 = opposite adjacent AC BC

To measure the height of a tree, Alma walked 125 ft from the tree and measured a 32° angle from the ground to the top of the tree. Estimate the height of the tree. The tree forms a right angle with the ground, so you can use the tangent ratio to estimate the height of the tree. tan 32° = height 125 height = 125 (tan 32°) height = 125 (0.624869351909) The tree is about 78 ft tall.

The ratio of the leg opposite to A to the hypotenus is fixed. B C Given A, The ratio of the leg opposite to A to the hypotenus is fixed. This ratio is called the sine. Sine A = Length of leg opposite A hypotenuse Sin A = opposite hypotenuse

The ratio of the leg adjacent to A to the hypontenus is fixed. B C Given A, The ratio of the leg adjacent to A to the hypontenus is fixed. This ratio is called the cosine. Cosine A = Length of leg adjacent A hypotenuse Cos A = adjacent hypotenuse

Sine and Cosine Ratios Use the triangle to find sin T, cos T, sin G, and cos G. Write your answer in simplest terms. cos G = = 12 20 3 5 adjacent hypotenuse sin G = = 16 20 4 5 opposite hypotenuse sin T = = 12 20 3 5 opposite hypotenuse cos T = = 16 20 4 5 adjacent hypotenuse

SOH SOHCAH SOHCAHTOA opposite hypotenuse adjacent hypotenuse adjacent B C Sin A = opposite hypotenuse Cos A = adjacent hypotenuse Tan A = opposite adjacent

Use your knowledge of special right triangles to find each trig ratio. sin 30° cos 30° tan 45° 30° 45°

Find the values of each trigonometric function sin 62° cos 38° tan 42° tan 38° sin 75° cos 25° cos 62° tan 5° sin 45°

Find the length BC. Round to the nearest hundredth. Find the length of QR. Round to the nearest hundredth. Find the length of FD. Round to the nearest hundredth.

HOMEWORK: Page 537, #20 – 34 (e) Page 545, #22 – 42 (e)