Download Right Triangle Unit Geometry 2 Vista Grande High School 1/24/2011

Right Triangle Unit
Geometry 2
Vista Grande High School
1/24/2011
Similarity Review
1/25/2011
1/26/2011
Similarity Test Chapter 8
1/27/2011
1/28/2011
Proportions in Right
Triangles &
Geometric Means
Pg. 532: 13 - 27
2/4/2011
1/31/2011
2/1/2011
2/2/2011
2/3/2011
Pythagorean Thm
Special Right
Trigonometric
and the Converse of
Triangles 30-60-90s
Ratios 9.5
Pythagorean Thm
and 45-45-90s 9.4
Pg. 562:10-33, 39, 41,
Pg. 538: 7 – 18, 31,34
42 (cont. next class)
Pg. 554: 1-20 +WS
Pg. 546: 8 - 25
2/7/2011
2/8/2011
2/9/2011
2/10/2011
2/11/2011 Pgs Rpts
Solving Right
Trig Ratios 9.5
Vectors 9.7
Pg. 562:10-33, 39, 41,
Triangles 9.6
Pg. 576: 10 - 20
42 (continued)
Pg. 570: 14-33, 39,40
2/14/2011
2/15/2011
2/16/2011
2/17/2011
2/18/2011
Right Triangles &
Right Triangles &
Tangents to a Circle
Trig Review
Trig Test (Chapter 9)
10.1
Materials available at spartansmath.wikispaces.com or spartansmathdolezal.wikispaces.com
Math Help Available: Black Lunch – Room 1218 (Ms. Wildermuth) or Gold Lunch – Room 1215 (Mrs. Hodge)
Term
Definition (in words)
Example (visual)
Converse of a Statement
Right Triangle
Proportion
Pythagorean Theorem
Hypotenuse
Legs of a Right Triangle
Altitude
Proportion
Ratio
Geometric Mean
Similar Right Triangles
C
Altitude from
Hypotenuse
Theorem
A
D
B
∆CDB ~ ∆ABC, ∆ACD ~ ∆ABC, and ∆CBD ~ ∆ACD
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.
B
12.3m
7.6 m
A
h
D
X
C
14.6 m
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
W
Y
5.1 ft
C
Z
Geometric Mean
Length of the
Altitude Theorem
A
D
9.5 ft
B
Example 2
x
6
Example 3
10
5
y
8
Example 5
Find the area of the triangle to the nearest tenth of a meter.
8m
h
10 m
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?
cable
50 ft
100 ft
100 ft
50 ft
8m
The Pythagorean Theorem
The
Pythagorean
Theorem
In a right triangle, the sum of the legs squared equals the
hypotenuse squared.
Pythagorean
Triple
When the sides of a right triangle are all integers it is called a
Pythagorean theorem.
c2 = a2 + b2, where a and b are legs and c is the hypotenuse.
3,4,5 make up a Pythagorean triple since
52 = 32 + 42.
Example 1
48
y
x
6
50
8
Example 2
q
p
90
50
100
90
Example 3
e
d
2
15
17
3
Example 4
g
f
5 3
5
4 3
8
c
a
b
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.
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.
Obtuse
Triangle
Theorem
c
a
b
If c2 = a2 + b2,
then ∆ABC is a right triangle
A
c
b
If c2 < a2 + b2, then ∆ABC is __________.
C
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
C
2
2
B
a
a
B
2
If c > a + b , then ∆ABC is __________.
Classifying
Triangles
Let c be the biggest side of a triangle, and a and b be the
other two side.
If c2 ____ a2 + b2, then the triangle is _________.
If c2 ____ a2 + b2, then the triangle is _________.
If c2 ____ a2 + b2, then the triangle is _________.
*** If a + b is not greater than c, a triangle cannot be
formed.
Determine what type of triangle, if any,
can be made from the given side
lengths.
7, 8, 12
11, 5, 9
5, 5, 5
1, 2, 3
16, 34, 30
9, 12, 15
13, 5, 7
13, 18, 22
4, 8, 4 3
5, 5 2, 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.
3.4 Special Right Triangles - Solve for each missing side. What pattern, if any do you notice?
3
2
3
2
5
4
5
4
7
6
7
6
½
300
½
300
45º-45º-90º
Triangles
Theorem
x
In a 45º-45º-90º triangle, the hypotenuse is
x 2
times each leg.
x
x
x
Solve for each missing length. What pattern, if any do you notice?
10
10
8
8
10
8
6
6
50
50
50
6
30º-60º-90º
Triangle Theorem
2x
2x
2x
In a 30º-60º-90º triangle, the hypotenuse is
____________ as long as the shortest leg, and the longer leg is
____________ times as long as the shorter leg.
30º
2x
x 3
60º
x
Example 1
Find each missing side length.
6
45º
45º
15
Example 2
18
12
30º
45º
Example 3
30º
44
12
30º
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?
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.
Main Idea
Details
Examples
B
sin =
F
6
Sine
3
10
8
I
6
Cosine
G
10
3A
6I
5
T
3O
12
A
5
M
M
10
13
A
4
D
Tangent
5
J
8
tan=
5
4
K
cos=
C
13T
12
R
5
N
13 Y
M
5
8
4
To determine the decimal
cos 23
sin 1
sin 17
Finding Trig
approximation of a trig value,
tanO32
Ecos 0 T
tanA65
Value
one may use a calculator or
sin 82
sin 82
cos 30
sin 90
tan 45
sin 30
Page 845.
What is an inverse?
Inverse of add:
Finding the
Measure of an
Angle
Inverse of multiply:
Inverse of subtract:
Inverse of divide:
Inverse of finding trig value:
G
12
T
Assume that G is an acute angle and tan G = 1.230. The
measure of G is _____.
Assume that R is an acute angle and sin R = 0.8910. The
measure of R is _____.
Assume that A is an acute angle and cos A = 0.3090. The
measure of A is _____.
Assume that N is an acute angle and tan N = 2.7475. The
measure of N is _____.
Assume that D is an acute angle and sin D = 0.1736. The
measure of D is _____.
You can solve a right triangle if
5
you hare…
20
x
15
Finding
1 side and 1 trig ratio.
Missing
1 side & 1 acute angle measure.
Measures
2 sides.
x
10
x
20
x
13
52
10
17
26
x
Find the missing angle and side measures of ABC given that
A = 60, C=90, and CB = 19.
Find the missing angle and side measures of RAT given that
R = 40, A=90, and RA = 14.
What if there is
no picture?
1. Draw the triangle.
2. Label measurements.
3. Determine appropriate trig
identity that should be used.
4. Solve.
Find the missing angle and side measures of DOG given that
D = 20, G=90, and DO = 20.
Two legs of a right triangle have lengths 15 and 8. The
measure of the smaller acute angle is _______.
Two legs of a right triangle have lengths 10 and 13. The
measure of the larger acute angle is _______.
To find the height of a tower, a surveyor positions a transit that
is 2 meter tall at a spot 80 meters from the base of the tower.
She measures the angle of elevation to the top of the tower to
be 62. What is the height of the tower, to the nearest meter?
Story
Problems
1.
2.
3.
4.
5.
Read the problem through.
Reread, looking for details.
Draw the situation.
Label measurements.
Determine appropriate trig
identity that should be used.
6. Solve.
A slide 5.6 meters long makes an angle of 24 with the ground.
How high is the top of the slide above the ground?
Liola drives 30 miles up a hill that is at a grade of 11. What
horizontal distance, to the nearest tenth of a mile, has she
covered?
Example 1
 Find each trigonometric ratio.
 sin A
A
 cos A
5
3
 tan A
 sin B
C
4
B
 cos B
 tan B
 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.
D
25
7
F
24
E
 Example 3
 Find the sine, cosine, and the tangent of A.
B
18√2
18
C
18
A
 Example 4
 Find the sine, cosine, and tangent of A.
B
10
5
C
5√3
A
 Example 5
 Use the table of trig values to approximate the sine, cosine, and tangent of 82°.
 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.
depression
 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.
 Example 7
A driveway rises 12 feet over a distance d at an angle of 3.5°. Estimate the length of the driveway.
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
 Example 1
 Find the value of each variable. Round decimals to the nearest tenth.
c
8
25
º
b
Example 2
 Find the value of each variable. Round decimals to the nearest tenth.
c
b
42º
40
 Example 3
 Find the value of each variable. Round decimals to the nearest tenth.
b
8
20º
a
 Example 4
 Find the value of each variable. Round decimals to the nearest tenth.
c
b
17º
10
 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.
h
B
A