Unit 3: Right Triangles and Trigonometry Geometry 2 Unit 3: Right Triangles and Trigonometry

3.1 Similar Right Triangles Geometry 2 3.1 Similar Right Triangles

Similar Right Triangles Altitude from Hypotenuse 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 each other. A C B D ∆CDB ~ ∆ACB, ∆ACD ~ ∆ABC, and ∆CBD ~ ∆ACD

Similar Right Triangles Example 1 A roof has a cross section that is a right triangle. The diagram shows the approximate dimensions of this cross section. Identify the similar triangles in the diagram. Find the height h of the roof. A D C B h 7.6 m 12.3m 14.6 m ∆ABD ~ ∆BCD ~ ∆ACB 6.6m

Similar Right Triangles Geometric Mean The geometric mean of two numbers a and b is the positive number x such that

Similar Right Triangles Geometric Mean Length of the 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. A C B D BD = CD CD AD

Similar Right Triangles Geometric Mean Length of Legs 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. AB = CB CB DB AB = AC AC AD

Similar Right Triangles Example 2 Find the value of x. 6 10 x 2√15

Similar Right Triangles Example 3 Find the value of y. 5 8 y √65

Similar Right Triangles Example 4 To estimate the height of a statue, your friend holds a cardboard square at eye level. She lines up the top edge of the square with the top of the statue and the bottom edge with the bottom of the statue. You measure the distance from the ground to your friends eye and the distance from your friend to the statue. In the diagram, XY = h – 5.1 is the difference between the statues height h and your friends eye level. Solve for h. h Z X 9.5 ft Y W 5.1 ft About 22.8 ft

3.2 The Pythagorean Theorem Geometry 2 3.2 The Pythagorean Theorem

The Pythagorean Theorem In a right triangle, the sum of the squares of the legs equals the square of the hypotenuse. c2 = a2 + b2, where a and b are legs and c is the hypotenuse. a c b

The Pythagorean Theorem Pythagorean Triple When the sides of a right triangle are all integers it is called a Pythagorean triple. 3,4,5 make up a Pythagorean triple since 52 = 32 + 42.

The Pythagorean Theorem Example 1 Find the unknown side lengths. Determine if the sides form a Pythagorean triple. 6 8 48 50 y x

The Pythagorean Theorem Example 2 Find the unknown side lengths. Determine if the sides form a Pythagorean triple. 100 q 90 50 p

The Pythagorean Theorem Example 3 Find the unknown side lengths. Determine if the sides form a Pythagorean triple. 2 3 e 17 15 d

The Pythagorean Theorem Example 4 Find the unknown side lengths. Determine if the sides form a Pythagorean triple. 5 g 8 f

The Pythagorean Theorem Example 5 Find the area of the triangle to the nearest tenth of a meter. 8 m 10 m h About 31.2 square meters

The Pythagorean Theorem Example 6 The two antennas shown in the diagram are supported by cables 100 feet in length. If the cables are attached to the antennas 50 feet from the ground, how far apart are the antennas? 50 ft 100 ft cable

3.3 The Converse of the Pythagorean Theorem Geometry 2 3.3 The Converse of the Pythagorean Theorem

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. a c b If c2 = a2 + b2, then ∆ABC is a right triangle.

a c2 = a2 + b2 b If a and b stay the same length and we make the angle between them smaller, what happens to c?

a c2 = a2 + b2 b If a and b stay the same length and we make the angle between them bigger, what happens to c?

Acute Triangle Theorem 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 acute. a c b A C B If c2 < a2 + b2, then ∆ABC is acute.

Obtuse Triangle Theorem 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 obtuse. a c b A C B If c2 > a2 + b2, then ∆ABC is acute.

Classifying Triangles Let c be the biggest side of a triangle, and a and b be the other two sides. If c2 = a2 + b2, then the triangle is right. If c2 < a2 + b2, then the triangle is acute. If c2 > a2 + b2, then the triangle is obtuse. *** If a + b is not greater than c, a triangle cannot be formed.

Example 1 Example 1 Determine what type of triangle, if any, can be made from the given side lengths. 7, 8, 12 11, 5, 9

Example 2 Example 2 Determine what type of triangle, if any, can be made from the given side lengths. 5, 5, 5 1, 2, 3

Example 3 Example 3 Determine what type of triangle, if any, can be made from the given side lengths. 16, 34, 30 9, 12, 15

Example 4 Example 4 Determine what type of triangle, if any, can be made from the given side lengths. 13, 5, 7 13, 18, 22

Example 5 Example 5 Determine what type of triangle, if any, can be made from the given side lengths. 4, 8, 5, , 5

Example 6 You want to make sure a wall of a room is rectangular. A friend measures the four sides to be 9 feet, 9 feet, 40 feet, and 40 feet. He says these measurements prove the wall is rectangular. Is he correct? You measure one of the diagonals to be 41 feet. Explain how you can use this measurement to tell whether the wall is rectangular. No

3.4 Special Right Triangles Geometry 2 3.4 Special Right Triangles

Special Right Triangles Solve for each missing side. What pattern, if any do you notice? 2 3

Special Right Triangles Solve for each missing side. What pattern, if any do you notice? 4 5

Special Right Triangles Solve for each missing side. What pattern, if any do you notice? 6 7

Special Right Triangles Solve for each missing side. What pattern, if any do you notice? 300 ½

Special Right Triangles Use the pattern you noticed on the previous page to find the length of the hypotenuse in terms of x. x

Special Right Triangles 45º-45º-90º Triangles Theorem In a 45º-45º-90º triangle, the hypotenuse is times each leg. x

Special Right Triangles Solve for each missing length. What pattern, if any do you notice? 10

Special Right Triangles Solve for each missing side. What pattern, if any do you notice? 8

Special Right Triangles Solve for each missing side. What pattern, if any do you notice? 6

Special Right Triangles Solve for each missing side. What pattern, if any do you notice? 50

Special Right Triangles 2x

Special Right Triangles 30º-60º-90º Triangle Theorem In a 30º-60º-90º triangle, the hypotenuse is twice as long as the shortest leg, and the longer leg is times as long as the shorter leg. 2x x 30º 60º

Special Right Triangles Example 1 Find each missing side length. 45º 6 45º 15

Special Right Triangles Example 2 45º 12 30º 18

Special Right Triangles Example 3 30º 44 30º 12

Special Right Triangles Example 4 A ramp is used to unload trucks. How high is the end of a 50 foot ramp when it is tipped by a 30° angle? By a 45° angle? 25 ft. About 35.4 ft

Special Right Triangles Example 5 The roof on a doghouse is shaped like an equilateral triangle with height 3 feet. Estimate the area of the cross-section of the roof. About 5.2 square feet

Geometry 2 3.5 Trigonometric Ratios

Trigonometric Ratios Name the side opposite angle A. Name the side adjacent to angle A. Name the hypotenuse. A C B

Trigonometric Ratios The 3 basic trig functions and their abbreviations are sine = sin cosine = cos tangent = tan

Trigonometric Ratios sin = opposite side SOH hypotenuse cos = adjacent side tan = opposite side adjacent side SOH CAH TOA

Trigonometric Ratios Example 1 Find each trigonometric ratio. sin A cos A tan A sin B cos B tan B 3 4 C A 5 B

Trigonometric Ratios Example 2 Find the sine, the cosine, and the tangent of the acute angles of the triangle. Express each value as a decimal rounded to four decimal places. 7 24 E D 25 F

Trigonometric Ratios Example 3 Find the sine, cosine, and the tangent of A. 18 18√2 C A B

Trigonometric Ratios Example 4 Find the sine, cosine, and tangent of A. 5 5√3 10 B A C

Trigonometric Ratios Example 5 Use the table of trig values to approximate the sine, cosine, and tangent of 82°.

Trigonometric Ratios Angle of Elevation When you stand and look up at a point in the distance, the angle that your line of sight makes with a line drawn horizontally is called the angle of elevation. angle of elevation angle of depression

Trigonometric Ratios Example 6 You are measuring the height of a building. You stand 100 feet from the base of the building. You measure the angle of elevation from a point on the ground to the top of the building to be 48°. Estimate the height of the building. About 111 feet

Trigonometric Ratios Example 7 A driveway rises 12 feet over a distance d at an angle of 3.5°. Estimate the length of the driveway. 197 ft

3.6 Solving Right Triangles Geometry 2 3.6 Solving Right Triangles

Solving Right Triangles Solving a Right Triangle To solve a right triangle means to determine the measures of all six parts. You can solve a right triangle if you know: Two side lengths One side length and one acute angle measure

Solving Right Triangles Example 1 Find the value of each variable. Round decimals to the nearest tenth. 25º 8 c b

Solving Right Triangles Example 2 Find the value of each variable. Round decimals to the nearest tenth. 42º 40 b c

Solving Right Triangles Example 3 Find the value of each variable. Round decimals to the nearest tenth. 20º a 8 b

Solving Right Triangles Example 4 Find the value of each variable. Round decimals to the nearest tenth. 17º 10 c b

Solving Right Triangles Example 5 During a flight, a hot air balloon is observed by two persons standing at points A and B as illustrated in the diagram. The angle of elevation of point A is 28°. Point A is 1.8 miles from the balloon as measured along the ground. What is the height h of the balloon? Point B is 2.8 miles from point A. Find the angle of elevation of point B. B A h 0.96 mi 43.8°