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Geometry Unit 7 Notes Packet Name:___________________ 7.1 Apply the Pythagorean Theorem 7.2 Use the Converse of the Pythagorean Theorem 7.3 Use Similar Right Triangles 7.4 Special Right Triangles 7.5 Apply the Tangent Ratio 7.6 Apply the Sine and Cosine Ratios 7.7 Solve Right Triangles 7.1: Theorem 7.1: Pythagorean Theorem: In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. Find the length of the hypotenuse of the right triangle: A 16 foot ladder rests against the side of a house, and the base of the ladder is 4 feet away. Approximately how high above the ground is the top of the ladder? Find the area of the isosceles triangle with following side lengths: a. 10 meters, 13 meters, and 13 meters. b. 30 feet, 18 feet, and 18 feet. c. 20 meters, 26 meters, and 26 meters. 1 Pythagorean Triples: A Pythagorean triple is a set of three positive integers, a, b, and c that satisfy the equation c2 = a2 + b2. 3,4,5 5,12,13 8,15,17 7,24,25 Find the length of the hypotenuse of the right triangle with side lengths of 24 and 10. Find the unknown side length of each triangle: 2 7.2: Theorem 7.2: 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 is a right triangle. Tell whether the given triangle is a right triangle: Tell whether a triangle with the given side lengths is a right triangle: a. 4,4 3 ,8 b. 10, 11, 14 c. 5,6, 61 Theorem 7.3: 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. Theorem 7.4: If the square of the length of the longest side of a triangle is greater than the sum of the squares of the lengths of the other two sides, then the triangle is an obtuse triangle. Can segments with lengths of 4.3 feet, 5.2 feet, and 6.1 feet form a triangle? If so, would the triangle be acute, right, or obtuse? 3 7.3: Theorem 7.5: 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. The diagram below shows a cross-section of a swimming pool. What is the maximum depth of the pool? Identify the similar triangles. Then find the value of x. 4 Proportions involving Geometric Means in Right triangle ABC: BD CD CD AD AB CB CB DB AB AC AC AD Find the value of y. Write your answer in simplest radical form. Theorem 7.6: Geometric Mean (Altitude) Theorem: 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 7.7: Geometric Mean (Leg) Theorem: 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. 5 7.4: Theorem 7.8: 45-45-90 Triangle Theorem: In a 45-45-90 triangle, the hypotenuse is as long as each leg. Hypotenuse = leg 2 . Find the length of the hypotenuse: Find the value of the variable: 6 2 times Theorem 7.9: 30-60-90 Triangle Theorem: In a 30-60-90 triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is 3 times as long as the shorter leg. Hypotenuse = 2 shorter leg Longer leg = shorter leg 3 The logo on a recycling bin resembles an equilateral triangle with side lengths of 6 cm. What is the approximate height of the logo? Find the values of x and y. Write your answer in simplest radical form. The body of a dump truck is raised to empty a load of sand. How high is the 14 foot body from the frame when it is tipped upward at the given angle? a. 45 degree angle b. 60 degree angle 7 7.5: A ________________________ ratio is a ratio of the lengths of two sides in a right triangle. The ratio of the lengths of the legs in a right triangle is constant for a given angle measure. This ratio is called the _______________ of the angle. Tan A = Find tan S and tan R. Write each answer as a fraction and as a decimal rounded to four places. Find tan J and tan K. Round to four decimal places. 8 Find the value of x: A roller coaster makes an angle of 52 degrees with the ground. The horizontal distance from the crest of the hill to the bottom of the hill is about 121 feet. Find the height, h, of the roller coaster to the nearest foot. A surveyor is standing 118 feet from the base of the Washington Monument. The surveyor measures the angle between the ground and the top of the monument to be 78 degrees. Find the height, h, of the Washington Monument to the nearest foot. 9 7.6: The _________ and ____________ ratios are trigonometric ratios for acute angles that involve the lengths of a leg and the hypotenuse of a right triangle. sin A = cos A = Find sin S, sin R, cos S, and cos R. Find cos U, cos W, sin U, sin W. 10 Angles: If you look up at an object, the angle your line of sight makes with a horizontal line is called the angle of ____________________. If you look down at an object, the angle your line of sight makes with a horizontal line is called the angle of _____________________. You are skiing on a mountain with an altitude of 1200 meters. The angle of depression is 21 degrees. About how far do you ski down the mountain? You want to build a skateboard ramp with a length of 14 feet and an angle of elevation of 26 degrees. You need to find the height and the length of the base of the ramp. 11 7.7: To solve a right triangle means to find the measures of all of its sides and angles. You can solve a right triangle if you know either of the following: __________ side lengths __________ side length and the measure of one ____________ angle Inverse Trigonometric Ratios: Inverse Tangent: If tan A = x, then tan-1 x = m<A Inverse Sine: If sin A = y, then sin-1 y = m<A Inverse Cosine: If cos A = z, then cos-1 z = m<A Use a calculator to approximate the measure of <A and <B to the nearest tenth of a degree: Let <A and <B be acute angles in a right triangle. Use a calculator to approximate the measure of <A and <B to the nearest tenth of a degree: a. sin A = 0.87 b. cos B = 0.15 c. sin D = 0.54 Solve the right triangle. Round your answer to the nearest tenth if necessary. 12