Unit 7: Right Triangles and Trigonometry Lesson 7.1 Use Inequalities in a Triangle Lesson 5.5 from textbook Objectives Use the triangle measurements to decide which side is longest and which angle is largest. Use the Triangle Inequality to determine the different possible side lengths of a triangle. Find the longest side and largest angle in a triangle. A 1. Use a ruler and find the measure the sides of ABC. 2. Use a protractor and find the measure of the angles of ABC. B C 3. Order the sides from shortest to longest.. 4. Order the angles from smallest to largest. 5. What conclusion can you draw about the longest side and largest angle in a triangle? How are the two connected? Triangle Inequality Theorem A) If one side of a triangle is than another side, then the the is larger than the the. B) If one angle of a triangle is than the other angle, then the the is longer than the the. Example 1 A) List the sides of the triangle in order from B) List the angles of the triangle in order from shortest to longest. largest to smallest.
Which side lengths make a triangle? The straws represent sides of a triangle. Use the straws to determine which combination of sides will and will not form a triangle. 8cm, 10cm, 15cm 4cm, 6cm, and 10cm 8cm, 4cm, 15cm 8cm, 4cm, 10 cm What conjecture can you make about the side lengths of a triangle? Triangle Inequality Theorem C) The sum of the lengths of any two sides of a triangle is greater than the length of the third side. JK + KL JL KL + > JK JL + > KL Example 2 Determine whether is it possible to construct a triangle with the given side lengths. Explain. A) 2, 5, 6 B) 4, 3, 9 Example 3 Describe the possible lengths of the third side of the triangle with the given lengths of the other two sides. 3 meters and 4 meters
Unit 7: Right Triangles and Trigonometry Lesson 7.2 Apply Pythagorean Theorem and Its Converse Objectives Use the Pythagorean Theorem to find the length of the unknown side of a right triangle and the area of the triangle. Determine which numbers form a Pythagorean Triple Use the Converse of the Pythagorean Theorem to determine whether the side lengths of the triangle form a right triangle. Classify the triangles as acute, right, or obtuse using the length of the sides and the Converse of the Pythagorean Theorem. RIGHT TRIANGLES Pythagorean Theorem In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of its legs. Example 1 Example 2 Find the length of the leg of the triangle. x = Find the area of the isosceles triangle with side lengths 10 meters, 13 meters, and 13 meters. A = PYTHAGOREAN TRIPLES Three positive integers a, b, and c that satisfy the equation c 2 = a 2 + b 2. Example 3 Complete the table of Pythagorean Triples and their multiples. 3, 4, 5 5, 12, 13 8, 15, 17 7, 24, 25 Multiple of 2 6, 8, 10 14, 28, 50 15, 36, 39 40, 75, 85 Multiple of x
Converse of the Pythagorean Theorem If the square of the length of the longest side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle. If c 2 = a 2 + b 2, then Example 3 ABC Tell whether the given triangle is a right triangle. Explain. A) B) Classifying Triangles Theorem #1 If the square of the length of the longest side of a triangle is less than the sum of the squares of the lengths of the other two sides, then the triangle is an acute triangle. If c 2 < a 2 + b 2, then Classifying Triangles Theorem #2 If the square of the length of the longest side of a triangle is more than the sum of the squares of the lengths of the other two sides, then the triangle is an obtuse triangle. If c 2 > a 2 + b 2, then Example 4 Can segments with lengths 4.3 feet, 5.2 feet, and 6.1 feet form a triangle? If so, would the triangle be acute, right, or obtuse?
Unit 7: Right Triangles and Trigonometry Lessons 7.3 Similar Right Triangles Objectives Solve problems involving similar right triangles formed by the altitude drawn to the hypotenuse of a right triangle. Use the geometric mean to solve problems. Right Triangle Similarity Theorem 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. CBD ~ ABC, ACD ~ ABC, CBD ~ ACD Example 1 Identify the similar triangles in the diagram. Write a similarity statement that relates the triangles. Similarity Statement Example 2 Identify the similar triangles. Find the value of x. Similarity Statement x = GEOMETRIC MEAN The geometric mean of two numbers a and b is the positive number x so that a x =. x b Example 2 Find the geometric mean of the numbers 5 and 7. Round your answer to the nearest tenth.
GEOMETRIC MEAN IN RIGHT TRIANGLES ABC with altitude CD forms two smaller triangles so that ABC ~ CBD ~ ACD. Proportions Involving Geometric Means in Right ABC. Length of shorter leg of I Length of shorter leg of II BD = CD CD AD Length of longer leg of I Length of longer leg of II CD is the geometric mean of BD and AD Length of hypotenuse of III Length of hypotenuse of I AB = CB CB DB Length of shorter leg of III Length of shorter leg of I CB is the geometric mean of AB and DB. Length of hypotenuse of III Length of hypotenuse of II AB = AC AC AD Length of longer leg of III Length of longer leg of II AC is the geometric mean of AB and AD. Example 3 Find the value of y. Write your answer in simplest radical form. Geometric Mean (Altitude) Theorem The length of the altitude of the right triangle is the geometric mean of the lengths of the two segments. BD CD =
Geometric Mean (Leg) Theorem The length of the altitude 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. AB AB = and = CB AC Example 4 Use the Geometric Mean Theorems to find AC and BD. AC = BD =
Unit 7: Right Triangles and Trigonometry Lesson 7.4: Special Right Triangles Objectives Use the 45-45-90 and 30-60-90 Triangle Theorems to find the lengths of sides of right triangles. Use the 45-45-90 and 30-60-90 Triangle Theorems to find the area of triangles. Isosceles Right Triangle Formed by cutting a square in half. If each leg has a length of x, what is the length of the hypotenuse? 45 o -45 o -90 o Triangle Theorem In a 45 o -45 o -90 o triangle, the hypotenuse is 2 times as long as each leg. Example 1 Find the length of the hypotenuse for each triangle. Example 2 Find the lengths of the legs in the triangle.
Equilateral Triangle If you cut the triangle in half, it forms a special right triangle. A 1) Use a ruler and draw a line that cuts the triangle in half. Label it AD. 2) Determine the measure of the following angles: m<adc = m<dac = m<dca = B C 3) If DC = x, use the Pythagorean and find AD and AC. AD = AC = 30 o -60 o -90 o Triangle In a 30 o -60 o -90 o triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is 3 times as long as the shorter leg. Example 3 Find the value of the variable. Write your answer in simplest radical form. h = x = y = Example 4 Complete the following table. d 5 e 8 3 12 f 14 18 3
Unit 7: Right Triangles and Trigonometry Lesson 7.5: Apply the Tangent Ratio Objectives Define basic trigonometric ratios in right triangles: tangent. Apply right triangle trigonometric ratios to solve problems involving missing lengths and angle measures in triangles. Vocabulary Trigonometric Ratio: *You will use trigonometric ratios to find the measure of a side or an acute angle in a right triangle. Tangent Ratio tan A = Length of leg opposite <A Length of leg adjacent to <A BC = AC tan B = Length of leg opposite <B Length of leg adjacent to <B AC = BC m<a + m<b = Example 1 Find tan S and tan R. Write each answer as a fraction and as a decimal rounded to four places. tan S = fraction decimal tan R = fraction decimal
Example 2 Find the value of x. x = Example 3 Find tan 60 o and tan 30 o using a special right triangle. tan 60 o = tan 30 o = Example 4 Find the area of the right triangle. Round your answer to the nearest tenth. A = Example 5 Find the perimeter of the right triangle. Round your answer to the nearest tenth. P =
Unit 7: Right Triangles and Trigonometry Lesson 7.6: Apply the Sine and Cosine Ratios Objectives Define basic trigonometric ratios in right triangles: sine and cosine. Apply right triangle trigonometric ratios to solve problems involving missing lengths and angle measures in triangles. Sine and Cosine Ratios sin A = Length of leg opposite <A Length of hypotenuse BC = AB cos A = Length of leg adjacent to <A Length of hypotenuse AC = AB Example 1 Find sin S and sin R. Find cos S and cos R. Round your answer to two four places. sin S = sin R = cos S = cos R = Example 2 You want to string a cable to make a dog run from two corners of a building, as shown in the diagram. Write and solve a proportion using a trigonometric ratio to approximate the length of the cables you need. x =
Angle of Elevation and Depression Angle of Elevation: Angle made when looking up at an object. Angle of Depression: Angle made when looking down at an object. Example 3 If you are skiing on a mountain with an altitude of 1200 meters and the angle of depression is 21 o, how far do you ski down the mountain? x = Example 4 Find the unknown side length. Then find sin X and cos X. Write each answer as a fraction in simplest form and as a decimal. Round to four decimal places, if necessary. XY = Sin X = fraction decimal Cos X = fraction decimal
Unit 7: Right Triangles and Trigonometry Lesson 7.7: Solve Right Triangles Objective Use the Pythagorean Theorem and trigonometric ratios to solve for missing angle measures and side lengths in right triangles. SOLVING RIGHT TRIANGLES Find measures of all sides and angles. This can be done if you know the following: Inverse Trigonometric Ratios Let <A be an acute angle. Inverse Tangent BC If tan A = x, then tan -1 x = A tan -1 AC = m<a EX) tan -1 3 = o 4 Inverse Sine BC If sin A = x, then sin -1 x = A sin -1 AB = m<a EX) sin -1 3 = o 5 Inverse Cosine AC If cos A = x, then cos -1 x = A cos -1 AB = m<a EX) cos -1 4 = o 5 Example 1 Use a calculator to approximate the measure of <A to the nearest tenth of a degree. m<a = m<a =
Example 2 Solve the right triangle. Round answers to the nearest tenth. m<k = ML = KL = m<d = m<f = EF = Law of Sines If ABC has sides of length a, b, and c as shown, then sin A sin B sin C = =. a b c Example 3 Use the information in the diagram to determine how much closer you live to the music store than your friend does. a = b = Law of Cosines If ABC has sides of length a, b, and c then, a 2 2 2 2 2 2 = b + c 2 b c cos A b = a + c 2 a c cos B c 2 = a 2 + b 2 2 a b cosc Example 4 In ABC at the right find m<c.