9 - wsfcs.k12.nc.us



9.1 Similar Right Triangles

If the altitude to the hypotenuse is drawn, then the two triangles formed are similar to each other and the original triangle.

C

ABC ~

BCD ~

ACD ~

A D B

AB BC CA AB BC CA

BC BD CD AC CD AD

There are two Geometric Means formed by the above figure

1. The altitude is the GM between the two parts of the hypotenuse.

AD CD C

CD BD

CD = √ AD x BD

A D B

2. Each leg is a GM between the hypotenuse and the adjacent part of the hypotenuse. C

AB AC AB BC

AC AD BC BD

AC = √AB x AD BC = √ AB x BD

A D B

Name 3 similar triangles and solve for x.

Solve for x, y, and z.

531: 1 – 30 all

9.2 The Pythagorean Theorem

EQ: What is the Pythagorean Theorem and what does it do?

In a right triangle, the short leg squared plus the long leg squared is equal to the hypotenuse squared.

a c a² + b² = c²

(short leg)²+ (long leg)² = (hypotenuse)²

b

Find the missing side of the right triangle.

538: 1 – 30

9.3 The Converse of the Pythagorean Theorem

EQ: What does the converse of the Pythagorean Theorem do?

a² + b² > c² a² + b² = c² a² + b² < c²

Acute triangle Right triangle Obtuse triangle

Each formula is (short leg) ² + (long leg) ² compared to (hypotenuse) ²

Classify each triangle as acute, right, or obtuse.

Can these lengths form a triangle? If so, classify each as an acute, a right, or an obtuse triangle.

6, 8, 9 5, 8, 10 2, 10, 12

13, 6, 7 4, 5, 5 16, 30, 34

545: 3 – 28 all

9.4 Special Right Triangles: finding sides without trig or Pythagoras

EQ: What are the special relationships in special right triangles?

45 -45 -90 Triangle 30 -60 -90 Triangle

Leg = leg short leg = ½ hypotenuse

Hypotenuse = (leg)√2 hypotenuse = 2(short leg)

Long leg = (short leg) √3

Find the missing sides of each triangle.

554: 1 – 23 all

9.5 Trigonometric Ratios

EQ: What are the three main trigonometric ratios found in right triangles?

Trigonometric ratio, or trig ratio, is a ratio of the lengths of the sides of a right triangle. These ratios only work in right triangles and the three basic ratios are sine, cosine, and tangent and are abbreviated as sin, cos, and tan.

Let ABC be a right triangle. The sin, cos, and tan will be defined as:

B

Sin = opposite side

Hypotenuse c a

Cos = adjacent side A C

hypotenuse b

sin A = cos A = tan A =

Tan = opposite side

adjacent side sin B = cos B = tan B =

Find the sine, cosine, and tangent of the acute angles.

Find the value of each variable. Round decimals to nearest tenth.

9.6 Solving Right Triangles

EQ: What does it mean so solve right triangles and how is it done?

Solving right triangles means you determine the measures of all six parts: three sides and three angles.

To solve a right triangle you must begin with:

1. two side lengths

2. one side length and one acute angle

Examples:

9.6 Solving Right Triangles

Solving right triangles means you determine the measures of all six parts: three sides and three angles.

To solve a right triangle you must begin with:

1. two side lengths

2. one side length and one acute angle

Examples:

MAD Quiz 9.2

Copy each problem. Write an equation and solve for the missing side.

1. 2.

3. 4.

MAD Quiz 9.3

Use the following triangle sides to write an equation and then determine if the triangle is acute, right, obtuse, or not a triangle.

1. 5”,7”, 9” 2. 15cm, 20cm, 25cm 3. 6m, 7m, 8m

4. 5.

MAD Quiz 9.4

Draw and label each figure. Find the missing sides of the special right triangles. You may write your answers with square roots or decimals.

1. 2.

5 x =

14 x =

Y =

Y =

3. 4.

x = 200

X = 22

Y =

Y =

MAD QUIZ 9.5

Sketch the triangles, write a trig equation, and solve for the missing sides.

1. 2. 3.

4. 5.

539: 31 -34

31. 32.

562: 3 – 8 all, 10 –26 even

3. 4/5 = 0.8 4. 3/5 = 0.6 5. 4/3 = 1.3333

6. 3/5 = 0.6 7. 4/5 = 0.8 8. ¾ = 0.75

10. sin R = 0.8491 12. sin X = 0.8321 14. sin G = 0.8944

cos R = 0.5283 cos X =0.5477 cos G = 0.4472

tan R = 1.6071 tan X = 1.5000 tan G = 2.0000

sin S = 0.5283 sin Y = 0.5477 sin H = 0.4472

cos S = 0.8491 cos Y = 0.8321 cos H = 0.8944

tan S = 0.6222 tan Y = 0.6667 tan H 0.5000

16. sin 48 = 0.7431 18. tan 81 = 6.3138 20. cos 70 = 0.3420

22. sin 78 = 0.9781 24. tan 23 = 0.4245 26. sin 56 = 0.8290

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