8-1 Similarity in Right Triangles

Transcription

1 8-1 Similarity in Right Triangles In this chapter about right triangles, you will be working with radicals, such as 19 and 2 5. radical is in simplest form when: 1. No perfect square factor other then 1 is under the radical sign. 2. No fraction is under the radical sign. 3. No fraction has a radical denominator. irections: Simplify lasswork: p.288 E # Note: Radicals should always be written in simplest form If a, b, and are positive numbers and a = b then is the geometric mean between a and b. Notice that by multiplying means/etremes 2 = ab and by taking the square root of each side = ab. asically, to find the geometric mean of two # s, multiply them and take the square root. Note: The geometric mean always falls between the two numbers. irections: Find the geometric mean between the two numbers and and and and and 24 Review: altitude, 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 to each other. Note: ~ ~ has a different measure in all 3 s.

2 The following two corollaries are true because of the similar s. orollary 1: When the altitude is drawn to the hypotenuse of a right triangle, the length of the altitude is the geometric mean between the segments of the hypotenuse. Y orollary 2: When the altitude is drawn to the hypotenuse of a right triangle, each is the geometric mean between the hypotenuse and the segment of the hypotenuse that is adjacent to that. X Z Eamples and informal statements of these two corollaries are given. or 1 X Y = Y Z piece of hypotenuse altitude altitude = other piece of hypotenuse or 2 or 2 For XY : For YZ : XZ XY XZ YZ = = XY X YZ Z hypotenuse = piece of hyp. adj. to irections: Eercises refer to the diagram at right. 12. If N = 8 and N = 16, find N. N 13. If N = 4 and N = 12, find N. 14. If N = 4 and N = 8, find. 15. If = 18 and = 12, find N. 16. If = 6 and N = 4, find N.

3 8-2 The Pythagorean Theorem Radicals: Eample: p.291 #5, 6, 9 You do: p.291 #7-8, Pythagorean Theorem: In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the s. Find the value of. Use diagram for #1-8. If in is a right angle, then Proof on p.290 a + b = c. b a c Reminders:. The diagonals of a rhombus are bisectors of each other.. The altitude drawn to the base of an isosceles triangle is to and bisects the base. Find the value of

4 Find the length of the diagonals of a square with perimeter The diagonals of a rhombus have lengths 18 and 24. Find the perimeter of the rhombus. 19. rectangle has diagonals of 5 cm and its width is 3 cm. Find the length of the rectangle. 20. The perimeter of a rhombus is 100 cm, and one diagonal is 48 cm long. Find the length of the other diagonal.

5 8-3 The onverse of the Pythagorean Theorem Review: Theorems learned T-8-1 ~ ~ or 1 or 2 Since ~ Since ~ = = The Pythagorean Theorem says: If is a right triangle, then a + b = c. The converse is also true: If a + b = c, then is a right triangle. Theorem: If the square of one side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle. c a b irections: If a triangle is formed with sides having the lengths given, it is a right triangle? 1. 4, 7, , 21, , 2, , 1.5, 1.7 triangle with sides 3, 4, and 5 is a right triangle because = 5. ny triangle with sides 3n, 4n, and 5n, n > 0,is also a right triangle because (3 n) + (4 n) = (5 n). Multiples of any three lengths that form a right triangle will also form right triangles. These groups of three lengths are called Pythagorean triples. If you use them, you can save time and effort. 3, 4, 5 5, 12, 13 8, 15, 17 7, 24, 25 6, 8, 10 10, 24, 26 16, 30, 34 14, 48, 50 9, 12, 15 15, 36, 39 24, 45, 51 21, 72,

6 2 These theorems say that by comparing c and a 2 + b 2, you can determine if a triangle is acute, right, or obtuse. Note: c is always largest side. If c = a + b, then If c > a + b, then If c < a + b, then is a right angle and is obtuse and is acute and is a right triangle. and is an obtuse triangle is an a acute triangle Theorem: If the square of the longest side of a triangle is greater than the sum of the squares of the other two sides, then the triangle is an obtuse triangle. Theorem: If the square of the longest side of a triangle is less than the sum of the squares of the other two sides, then the triangle is an acute triangle. c a b a c b c a b Note: Remember to check if it is a. irections: If a triangle is formed with the given lengths, is it acute, right, or obtuse? 5. 8, 9, , 5, , 13, , 7, , 2 3, , 11, , 5, , 5, 5 3

7 8.4 Special Right Triangles Two special Types: I Solve: II () Solve: 3 2( ) III. Solve for

8 8-4 Special Right ngles (ay 2) So why does it work I hyp. 45 II long 30 hyp short III Solve:

9 IV () Solve: 3 2()

10 8.5 The Tangent Ratio Trigonometric Functions S - Sine O opp H hyp osine adj H hyp T Tangent O opp - adj opp. adj. adj. opp. hyp. hyp. Today we deal with Tangent(TO) I. Epress tan X and tan Y as ratios. 1. Z 2. Y X Z X Y Y X Z Y Z 20 X Refer to the Table of Trigonometric Ratios on pg. 311 of your tetbook. 5. tan tan tan tan tan tan

11 II. Find the value of to the nearest tenth freeway ramp has a 10% grade. What 17 angle does the ramp have with the ground? y 8 rise 200 [Note: Grade(%) = *100 ] run 60 z

12 8.6 The Sine and osine Ratios Trigonometric Functions S - Sine O opp H hyp osine adj H hyp T Tangent O opp - adj opp. adj. Today we deal with SOH and H adj. opp. hyp. hyp. I. Epress Sin, Sin, os and os as ratios

13 II. Find the value of and y to the nearest tenth y y 15. y y n n n 5

14 8-7 pplications of Right Triangles I. Terms To Know: ngle of Elevation ngle of epression II. The Problems 1. From a point 80m from the base of a tower, the of elevation to the top of the tower is 28 o. How tall is the tower? 2. ladder that is 20ft. is leaning against the side of a building. If the formed between the ladder and the ground is 75 o, how far is the bottom of the ladder from the base of the building? 3. When the sun is 62 o above the horizon, a building casts a shadow 18m long. How tall is the building? 4. kite is flying at an of elevation of about 55 o. Ignoring the sag in the string, find the height of the kite if 85m of string have been let out.

15 5. wire is attached to the top of a tower and to a point on the ground that is 35m from the base of the tower. If the wire makes a 65 o angle with the ground, how long is the wire? 6. The angle of depression from the top of a tower to a boulder on the ground is 38 o. If the tower is 25m high, how far from the base of tower is the boulder? 7. n observer at the top of a building sees a car on the road below. The of depression to the car is 28 o. If the car is about 50m from the building when it is seen, how tall is the building?