How To Solve The Pythagorean Triangle


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1 Name Period CHAPTER 9 Right Triangles and Trigonometry Section 9.1 Similar right Triangles Objectives: Solve problems involving similar right triangles. Use a geometric mean to solve problems. Ex. 1 Use the diagrams at the right to review definitions of a right triangle. 1. Name the legs of ABC. 2. Name the hyotenuse of ABC. 3. What is the measure of A and C? 4. Name the legs of DEF. 5. Name the hypotenuse of DEF. 6. Could DEF have an obtuse angle? Explain. Theorem 9.1 If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other. Ex. 2 Use the diagram. a. Sketch the three similar triangles. Label the vertices. b. Write similarity statements for the three triangles. Remember that the geometric mean of two positive numbers a and b is the positive number x such that a x x When you write statements of proportionality, some sides appear in more than one triangle. b Shorter leg of CBD Shorter leg of ACD Shorter leg of ABC Shorter leg of CBD Shorter leg of ABC Shorter leg of ACD BD CD CD AD BD CD CD AD BD CD CD AD longer leg of CBD longer leg of ACD longer leg of ABC longer leg of CBD longer leg of ABC longer leg of ACD The results are listed in the theorems on the next page. X 1
2 Theorem 9.2 In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments. The length of the altitude is the geometric mean of the lengths of the two segments. Theorem 9.3 In a right triangle, the altitude from the right angle to the hypotenuse Divides the hypotenuse into two segments. The length of each leg of the right triangle is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse that is adjacent to the leg. Ex. 3 Complete and solve the proportion x x BD CD CD AD AB CB CB DB AB AC AC AD 1. x x x Ex. 4 Write similarity statements for the three similar triangles in the diagram. Then find the given length. 4. Find QS. 5. Find TU. Ex. 5 Find the value of each variable
3 Section 9.2 The Pythagorean Theorem Objectives: Proving the Pythagorean Theorem Using the Pythagorean Theorem In this lesson, you will study one of the most famous theorems in mathematics the Pythagorean theorem. The relationship it describes has been known for thousands of years. No one know at what point in history this relationship was first discovered. The ancient Babylonians and Chinese were known to use this relationship. THEOREM 9.4 Pythagorean Theorem: There are many different proofs. Here is a visual one. A Pythagorean triple is Find the length of the hypotenuse of the right triangle. Tell whether the sides form a Pythagorean triple. Ex.1 Find the length of the leg of the right triangle. Ex. 2 3
4 Ex 3 Two boats start at the same point on a lake. One heads north for 100 yards and the other heads west for 50 yards. How far apart are the boats? Make a sketch first. Round to the nearest tenth. Ex. 4 Find the area of the triangle to the nearest tenth of a meter. Section 9.3 The Converse of the Pythagorean Theorem THEOREM 9.5 Converse of the Pythagorean: Ex. 5 The triangle appears to be a right triangle. Is it a right triangle? Theorem 9.4 Theorem 9.5 Theorem 9.6 If ABC is a right triangle, then c² = a² + b². If c² = a²+ b², then ABC is a right triangle. (Converse theorem) If c² < a² + b², then ABC is an acute triangle. Theorem 9.7 If c² > a² + b², then ABC is an obtuse triangle. Decide whether the set of numbers can represent the side lengths of a triangle. If they can, classify the triangle as right, acute, or obtuse. a. 1, 3, 5 b. 3, 5, 7 c. 10, 49, 50 d. 17, 144, 145 4
5 Objectives: Section 9.4 Special Right Triangles Side lengths of Special Right Triangles Special Right Triangles in real life. Theorem Triangle Theorem: Theorem Triangle Theorem: Right triangl special right triangles. If you are given one side, you can use the formulas to find the other two sides. They can be proven by using the Pythagorean Theorem. Find the value of each variable. Ex. 1 Ex. 2 Ex. 3 (Ex 3 in the book) Find the values of s and t. 5
6 Ex 4 Use the diagram to tell whether the equation is true or false. t 3h 7 h 2 Ex 5 a. The side lengths of an equilateral triangle is 5 cm. Find the length of an altitude of the triangle. (Hint: Draw a sketch first.) b. The perimeter of a square is 32 inches. Find the length of a diagonal. Ex 6 A tipping platform is a ramp used to unload trucks. How high is the end of an 80 foot ramp when Ex. 7 Given the following information, what are the measures of the 2 acute angles? Ex. 8 Find the area. Round to the nearest tenth. a. b. 6
7 Objectives: Section 9.5 Trigonometric Ratios Finding Trigonometric Ratio Using trigonometric Ratios in Real Life A trigonometric ratio is a The word trigonometry is derived from the ancient Greek language and means measurement of triangles. The three basic trigonometric ratios are sine, cosine. and tangent, which are abbreviated as sin, cos, and tan. respectively. The saying sohcahtoa helps remember the ratios. trigonometric ratio Let ABC be a right triangle. The sine, the cosine, and the tangent of the Acute angle A are defined as fallows. side opposite A sin A a hypotenuse c side adjacent A cos A b hypotenuse c side opposite A tan A side adjacent A a b Use the diagrams at the right to find the trigonometric ratio. 1. sin A 2. cos A 3. tan B 4. sin J 5. cos K 6. tan K Find the sine, the cosine, and the tangent of the acute angles of the triangle. Express each value as a decimal rounded to four places
8 Use a calculator to approximate the given value to four decimal places. Set your calculator to degrees. Find the value of each variable. Round decimals to the nearest tenth A train is traveling up a slight grade w traveling 1 mile what is the vertical change in feet? 19 horizontally about 58.2 meters. Estimate the height h of the slide. 8
9 Objectives: Section 9.6 Solving Right Triangles Solving a right triangle Using right triangles in real life sohcahtoa Match the trigonometric expression with the correct ratio. Some ratios may be used more than once, and some may not be used at all. 1. sin A = 2. cos A = 3. tan A = 4. sin B = 5. cos B = 6. tan B = Once you know the sine, the cosine, or the tangent of an acute angle, you can use a calculator to find the measure of the angle. In general, for an acute angle A: if sin A = x, then sin ¹x = m A. if cos A = x, then cos ¹x = m A if tan A = x, than tan ¹x = m A A is an acute angle. Use a calculator to approximate the measure of A. Round to one decimal place. 7. sin A = tan A = cos A = sin A = 0.38 Every right triangle has one right angle, two acute angles, one hypotenuse, and two legs. To solve a right triangle means to determine the measures of all six parts. You can solve a right triangle if you know either of the following: Two side lengths One side length and one acute angle measure Solve the right triangle. Round decimals to the nearest tenth
10 13. A sonar operator on a ship detects a submarine at a distance of 400 meters and an angle of 14. The Uniform Federal Accessibility Standards specify that the ramp angle used for a wheelchair vertical rise is E z e Section 9.7 Vectors Objective: Finding the Magnitude of a Vector Match the vector with the correct component form of the vector A. 2,4 B. 4,2 C. 2, 4 uuu r The magnitude of a vector AB is the distance from the initial point A to the terminal point B and uuu r is written AB. Write the vector in component form. Find the magnitude of the vector. Round your answer tot he nearest tenth
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