Download Unit 7: Right Triangles and Trigonometry Lesson 7.1 Use

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.
A
Find the longest side and largest angle in a triangle.
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
shortest to longest.
B) List the angles of the triangle in order from
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 __________________
8cm, 4cm, 15cm ____________________
4cm, 6cm, and 10cm __________________
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.
Find the area of the isosceles triangle with side
lengths 10 meters, 13 meters, and 13 meters.
x = ___________
A = __________________
PYTHAGOREAN TRIPLES
Three positive integers a, b, and c that satisfy the equation c2 = a2 + b2.
Example 3
Complete the table of Pythagorean Triples and their multiples.
Multiple of 2
3, 4, 5
6, 8, 10
5, 12, 13
8, 15, 17
15, 36, 39
40, 75, 85
Multiple of x
7, 24, 25
14, 28, 50
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 c2 = a2 + b2, then ∆ABC _________________________
Example 3
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 c2 < a2 + b2, 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 c2 > a2 + b2, 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
= ________
CB
and
AB
= ________
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?
_____________________________________
45o-45o-90o Triangle Theorem
In a 45o-45o-90o 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
A
If you cut the triangle in half, it forms a special right triangle.
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<DCA = ______________
C
B
m<DAC = __________
3) If DC = x, use the Pythagorean and find AD and AC.
AD = ___________
AC = _______________
30o-60o-90o Triangle
In a 30o-60o-90o 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
f
12
8 3
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 = BC
AC
Length of leg adjacent to <A
tan B =
Length of leg opposite <B
AC
Length of leg adjacent to <B =
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.
Example 3
Find tan 60o and tan 30o using a special right triangle.
tan 60o = _____________
tan 30o = _____________
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 = __________________________
x = _________________
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
BC
AB
sin A =
Length of leg opposite <A
Length of hypotenuse
cos A =
Length of leg adjacent to <A
AC
=
Length of hypotenuse
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 21o,
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
If tan A = x, then tan-1 x = A
Inverse Sine
If sin A = x, then sin-1x = A
Inverse Cosine
If cos A = x, then cos-1x = A
 BC 
tan-1  AC  = m<A
3
EX) tan-1   = _____________o
4
 BC 
sin-1  AB  = m<A
3
EX) sin-1   = _____________o
5
 AC 
4
cos-1  AB  = m<A EX) cos-1   = _____________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 = ______________
m<D = _______________
ML = ______________
m<F = _______________
KL = ______________
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 = b 2 + c 2 − 2 ⋅ b ⋅ c ⋅ cos A
c 2 = a 2 + b 2 − 2 ⋅ a ⋅ b ⋅ cos C
Example 4
In ∆ABC at the right find m<C.
b 2 = a 2 + c 2 − 2 ⋅ a ⋅ c ⋅ cos B