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Right Triangles and Trigonometry
Chapter 7 – Right Triangles & Trigonometry
Big IDEAS:
1) Using Pythagorean Theorem and its converse.
2) Using special relationships in right triangles.
3) Using trigonometric ratios to solve right triangles.
Section:
7 – 1 Applying the Pythagorean Theorem
Essential Question
If you know the lengths of two sides of a right triangle, how do you find the length of the third side?
Warm Up:
Key Vocab:
Pythagorean Triple
A set of three positive integers a, b, and c that satisfy the equation
The most common are:
3, 4, 5
5, 12, 13
8, 15, 17
7, 24, 25
Theorems:
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.
(acb)
Show:
Ex 1:Find the length of the hypotenuse of a right triangle with legs measuring 5 and 12.
The hypotenuse is 13 units.
Ex 2:Randy made a ramp for his dog to get into his truck. The ramp is 6 feet long and the bed of the truck is 3 feet above the ground. Approximately how far from the back of the truck does the ramp touch the ground?
(36)
Ex 3:Find the area of an isosceles triangle with side lengths 20 inches, 20 inches and 24 inches.
(20202412)
Ex 4:The given lengths are two sides of a right triangle. All three side lengths of the triangle are integers and together form a Pythagorean triple. Find the length of the third side and tell whether it is a leg or the hypotenuse.
a. 12, 13
5 – leg
b. 21, 72
75 – hypotenuse
c. 9, 15
12 – leg
Section:
7 – 2 Use the Converse of the Pythagorean Theorem
Essential Question
How can you use the sides of a triangle to be used to classify a triangle by its angles?
Warm Up:
Theorems:
Converse of the Pythagorean Theorem
(bca)
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 two shorter sides,
then
the triangle is a right triangle.
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 two shorter sides,
then
the triangle is an acute triangle.
If
the square of the length of the longest side of a triangle is greater than to the sum of the squares of the length of the two shorter sides,
then
the triangle is an obtuse triangle.
Show:
Ex 1: Tell if the given triangle is right, acute, or obtuse – sides are 11, 20, 23.
Since , the triangle is obtuse.
Ex 2:Tell if the given triangle is right, acute, or obtuse – sides are .
Since , the triangle is right.
Ex 3:The sides of a triangle have lengths x, x – 8, 40. If the length of the longest side is 40, what values of x will make the triangle acute?
Closure:
· How can you use the sides of a triangle to be used to classify a triangle by its angles?
If , then it is a right triangle.
If , then it is an obtuse triangle.
If , then it is an acute triangle.
Section:
7 – 3 Use Similar Right Triangles
Essential Question
How can you find the length of the altitude to the hypotenuse of a right triangle?
Warm Up:
Key Vocab:
Geometric Mean
For two positive numbers a & b, the positive number x that satisfiesSo, and .
Example:
Theorems:
(BADC)
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.
Geometric Mean (Altitude) Theorem
( CBAD)
If
the altitude is drawn to the hypotenuse of a right triangle,
Then
The length of the altitude is the geometric mean of the lengths of the two segments of the hypotenuse.
Geometric Mean (Leg) Theorem
( CBAD)
If
the altitude is drawn to the hypotenuse of a right triangle,
Then
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.
Show:
(BATS)Ex 1:Identify the similar triangles in the diagram.
Ex 2:Find the value of k.
(Geometric Mean Leg Theorem)
(k 210)
Ex 3:Find the value of k.
(Geometric Mean Altitude Theorem) ( 16)
(9)
(k)
Try: pg 448 #13 – 15
13. Geometric Mean Leg Theorem:
14. Geometric Mean Altitude Theorem:
15. Geometric Mean Leg Theorem:
Closure:
· Describe the transformation(s) that move the two smaller triangles onto each other.
The smaller triangles can be rotated onto each other
· Describe the transformation(s) that move either of the two smaller triangles onto the whole triangle.
The smaller triangles need to be reflected and then rotated onto the whole triangle.
Section:
7 – 4 Special Right Triangles
Essential Question
How do you find the lengths of the sides of a triangle and a triangle?
Warm Up:
Theorems:
Triangle
The hypotenuse is times as long as each leg.
Easy as … !
(xx)
Triangle
The hypotenuse is 2 times as long as the short leg.
The longer leg is times as long as the shorter leg.
Easy as … !
(2x1x)
Show:
Ex 1:Find the length of the hypotenuse.
a.
(12)
b.
(1010)
Ex 2:Find the length of the legs in the triangle.
(xx)
Ex 3:Alex has a team logo patch in the shape of an equilateral triangle. If the sides are 2.5 inches long, what is the approximate height of his patch?
(2x1x)
Ex 4:Find the values of a and b. Write your answers in simplest radical form.
(b 15a)
Ex 6: A kite is attached to a 100 foot string as shown. How far above the ground is the kite when the string forms the given angle?
a.
b.
Section:
7 – 5 Apply the Tangent Ratio
Essential Question
How can you find a leg of a right triangle when you know the other leg and one acute angle?
Warm Up:
Key Vocab:
Trigonometric Ratio
The ratio of the lengths of two sides in a right triangle.
Three common trigonometric ratios are sine, cosine, and tangent.
Tangent Ratio
Let , be a right triangle with acute angle , then
(ABChypotenuseleg adjacent to Aleg opposite of A)
Show:
Ex 1:Find the and the. Write each answer as a fraction and as a decimal rounded to four places.
( 60DEF4575)
Ex 2:Find the value of x.
(9x)
Ex 3:Find the height of the flagpole to the nearest foot.
(21)Ex 4:What is the exact value of the tangent ratio of a angle?
Section:
7 – 6 Apply the Sine and Cosine Ratios
Essential Question
How can you find the lengths of the sides of a right triangle when you are given the length of the hypotenuse and one acute angle?
Warm Up:
Key Vocab:
Angle of Elevation (Incline)
The angle of sight when looking up at an object
Angle of Depression (Decline)
The angle of sight when looking down at an object
(Angle of Elevation (Incline)) (Angle of Depression (Decline))
Key Concepts:
(ABChypotenuseleg adjacent to Aleg opposite of A)
Sine Ratio
Let , be a right triangle with acute angle , then
Cosine Ratio
Let , be a right triangle with acute angle , then
The great Chief of the Trigonometry Tribe SOH CAH TOA is a good memory device …
Another memory device … hippos
Show:
Ex 1:Find the and . Write each answer as a fraction and decimal rounded to four places.
(A408575)
Ex 2:Find the and . Write each answer as a fraction and decimal rounded to four places.
(R1510)
Ex 3:A rope, staked 20 feet from the base of a building, goes to the roof and forms an angle of elevation of . To the nearest tenth of a foot, how long is the rope?
Ex 4:A pilot is looking at an airport form her plane. The angle of depression is . If the plane is at an altitude of 10,000 ft, approximately how far is the air distance to the runway?
Ex 5:A dog is looking at a squirrel at the top of a tree. The distance between the two animals is 55 feet and the angle of elevation is . How high is the squirrel and how far is the dog from the base if the tree?
Ex 6: Use a special right triangle to find the sine and cosine of a angle.
(11)
Section:
7 – 7 Solve Right Triangles
Essential Question
In a right triangle, how can you find all the sides and angles of the triangle?
Warm Up:
Key Vocab:
Solve a Right Triangle
Find the lengths of all sides and the measures of all angles
Inverse Function
“Opposite Functions”
Functions that cancel each other out
Example:
Key Concepts:
Inverse Sine
This inverse function is used to find the measure of an acute angle in a right triangle when the opposite side and hypotenuse of the triangle are either given or can be determined.
“The angle whose sine value is…”
Notation:
How it works:if
(ABC)
Inverse Cosine
This inverse function is used to find the measure of an acute angle in a right triangle when the adjacent side and hypotenuse of the triangle are either given or can be determined.
“The angle whose cosine value is…”
Notation:
How it works: if
(ABC)
Inverse Tangent
This inverse function is used to find the measure of an acute angle in a right triangle when the opposite and adjacent sides of the triangle are either given or can be determined.
“The angle whose tangent value is…”
Notation:
How it works: if
(ABC)
Caution!
The notations are different from a negative exponent meaning to take a reciprocal. Example: Non-example:
Show:
Ex 1:Solve the triangle formed by the water slide shown in the figure. Round decimal answers to the nearest tenth. Note: there are multiple, correct solution paths.
Ex 2:Let be an acute angles in right triangles. Use a calculator to approximate the measure of each to the nearest tenth of a degree. Show your work.
a.
b.
c.
Ex 3:Use the calculator to approximate the measure of to the nearest tenth of a degree.
Ex 4:A road rises 10 feet in a horizontal distance of 200 feet. What is the angle of inclination?
( 200 ft 10 ftA B C)
Closure:
· When do you use Sine, Cosine, and Tangent?
Sine, cosine, and tangent are used to find a missing side length in a right triangle
· When do you use Inverse Sine, Inverse Cosine, and Inverse Tangent?
Inverse Sine, Inverse Cosine, and Inverse Tangent are used to find a missing angle measure in a right triangle.
Section:
7 - 8 Law of Sines and Law of Cosines
Essential Question
How do you find sides and angles of oblique triangles?
Warm Up:
Key Vocab:
Oblique Triangles
Triangles that have no right angles – they are either acute or obtuse.
Law of Sines
OR
(B)
(bac)
(C) (A)
Oblique , where
and side a are always opposite each other,
and side b are always opposite each other, and
and side c are always opposite each other
Law of Cosines
OR
OR
Key Concept:
Solving Oblique Triangles
Given Information
Solution Method
Two angles and any side: AAS or ASA
Law of Sines
* The Ambiguous Case * :
Two sides and a non-included angle: *SSA*
· may have one solution, two solutions, or no solutions.
Law of Sines
Three sides: SSS
Law of Cosines
Two sides and an included angle: SAS
Law of Cosines
Show:
Ex 1:Use the information in the diagram to find the distance each person lives from the school – round answers to the nearest hundredth. Show your work. How much closer to school does Jimmy live compared to Adolph?
(JimmyAdolphSchoolxyxx3 mi)
Jimmy is 1.78 miles closer to school than Adolph.
Ex 2:In , , and . Find f to the nearest hundredth.
(D)
(E) (F) (12) (9)
Ex 3:Solve if , and . Round all answers to the nearest tenth.
Case #1:
Case #2:
Closure:
· When do you apply the Law of Sines?
Use the Law of Sines when you are given two angles and a side or when you are given two sides and a non-included angle.
· When do you apply the Law of Cosines?
Use the Law of Cosines when you are given three sides or two sides and the included angle.
Student Notes Geometry Chapter 7 – Right Triangles and Trigonometry KEYPage #7
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xab
=
4
16
=
x
x
416
8
=×
=
x
x
~~
AB
CBD
A
C
CD
D
D
D
=
D
CD
B
CD
AD
Hyp. Seg.
Hyp. Seg.
Alt.
Alt.
=
B
B
BC
A
BC
D
=
Hyp.
Adj. Hyp
Leg
Leg
. Seg.
=
B
D
AC
A
AC
A
=
Hyp.
Adj. Hyp
Leg
Leg
. Seg.
=
~~
ABTASBBST
DDD
2
2
10
2
20
20
45
25
k
k
k
k
k
k
=
=
=
=
=
2
144
16
9
12
k
k
k
k
=
=
=
9
5
x
x
=
3
5
x
=
1218
18
y
=
27
y
=
27
2716
z
=
729
16
45.6
z
=
»
306090
--
ooo
454590
--
ooo
--
ooo
454590