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![Page 1: 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 a 2 + b 2 = c 2 a, leg.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649dd85503460f94acd798/html5/thumbnails/1.jpg)
CHAPTER 7
![Page 2: 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 a 2 + b 2 = c 2 a, leg.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649dd85503460f94acd798/html5/thumbnails/2.jpg)
7.1 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
a2 + b2 = c2
a, leg
b, leg
c, hypotenuse
![Page 3: 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 a 2 + b 2 = c 2 a, leg.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649dd85503460f94acd798/html5/thumbnails/3.jpg)
PYTHAGOREAN TRIPLE a set of 3 positive integers a, b, and c
that satisfy the equation a2 + b2 = c2.
3,4,55,12,13 8,15,17 7,24,25and multiples of these numbers like…. 6,8,10 10,24,26 16,30,34
14,48,50
![Page 4: 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 a 2 + b 2 = c 2 a, leg.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649dd85503460f94acd798/html5/thumbnails/4.jpg)
EXAMPLES: FIND X.
x
12
5
![Page 5: 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 a 2 + b 2 = c 2 a, leg.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649dd85503460f94acd798/html5/thumbnails/5.jpg)
SOLVE FOR X.
x
2 14
2 5
![Page 6: 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 a 2 + b 2 = c 2 a, leg.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649dd85503460f94acd798/html5/thumbnails/6.jpg)
A ramp for a truck is 6 feet long. The bed of the truck is 3 feet above the ground. How long is the base of the ramp?
![Page 7: 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 a 2 + b 2 = c 2 a, leg.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649dd85503460f94acd798/html5/thumbnails/7.jpg)
FIND THE AREA OF THE TRIANGLE
24in
20in. 20in.
![Page 8: 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 a 2 + b 2 = c 2 a, leg.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649dd85503460f94acd798/html5/thumbnails/8.jpg)
PROVING THE PYTHAGOREAN THEOREM
b
b
b
b
a
a
a
a c
c
c
c
![Page 9: 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 a 2 + b 2 = c 2 a, leg.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649dd85503460f94acd798/html5/thumbnails/9.jpg)
7.2 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.
If c2 = a2 + b2, then triangle ABC is a right triangle.
![Page 10: 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 a 2 + b 2 = c 2 a, leg.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649dd85503460f94acd798/html5/thumbnails/10.jpg)
OTHER THEOREMS FROM 7.2 If c2 < a2 + b2 , then the triangle is
acute.
If c2 > a2 + b2 , then the triangle is obtuse.
![Page 11: 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 a 2 + b 2 = c 2 a, leg.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649dd85503460f94acd798/html5/thumbnails/11.jpg)
PROVE: IF C2 < A2 + B2 , THEN THE TRIANGLE IS ACUTE
cb
a
A
C B
Given Diagram:
xb
a
P
R Q
![Page 12: 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 a 2 + b 2 = c 2 a, leg.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649dd85503460f94acd798/html5/thumbnails/12.jpg)
PROVE: IF C2 > A2 + B2 , THEN THE TRIANGLE IS OBTUSE. Given Diagram:
cb
a
A
C B
xb
a
P
R Q
![Page 13: 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 a 2 + b 2 = c 2 a, leg.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649dd85503460f94acd798/html5/thumbnails/13.jpg)
7.3 USING SIMILAR RIGHT TRIANGLES 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.
If CD is an altitude of ABC, then CBD ~ ABC ~ ACD
A
C
BD
![Page 14: 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 a 2 + b 2 = c 2 a, leg.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649dd85503460f94acd798/html5/thumbnails/14.jpg)
EXAMPLE: Identify the similar triangles
D
EF
G
![Page 15: 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 a 2 + b 2 = c 2 a, leg.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649dd85503460f94acd798/html5/thumbnails/15.jpg)
EXAMPLE: Side view of a tool shed What is the maximum height of the
shed to the nearest tenth?
17ft
9ft
15ft8ft
![Page 16: 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 a 2 + b 2 = c 2 a, leg.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649dd85503460f94acd798/html5/thumbnails/16.jpg)
GEOMETRIC MEAN (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 length of the two segments
A
C
BD
![Page 17: 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 a 2 + b 2 = c 2 a, leg.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649dd85503460f94acd798/html5/thumbnails/17.jpg)
GEOMETRIC MEAN (LEG) 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.
A
C
BD
![Page 18: 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 a 2 + b 2 = c 2 a, leg.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649dd85503460f94acd798/html5/thumbnails/18.jpg)
EXAMPLE: Find K
k
102
![Page 19: 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 a 2 + b 2 = c 2 a, leg.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649dd85503460f94acd798/html5/thumbnails/19.jpg)
7.4 SPECIAL RIGHT TRIANLGES 45o-45o-90o Triangle Theorem-
In a 45o-45o-90o triangle, the hypotenuse is √2 times as long as each leg.
![Page 20: 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 a 2 + b 2 = c 2 a, leg.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649dd85503460f94acd798/html5/thumbnails/20.jpg)
EXAMPLE: FIND X.
![Page 21: 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 a 2 + b 2 = c 2 a, leg.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649dd85503460f94acd798/html5/thumbnails/21.jpg)
EXAMPLE: FIND X.
![Page 22: 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 a 2 + b 2 = c 2 a, leg.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649dd85503460f94acd798/html5/thumbnails/22.jpg)
30O-60O-90O TRIANGLE THEOREM 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.
![Page 23: 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 a 2 + b 2 = c 2 a, leg.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649dd85503460f94acd798/html5/thumbnails/23.jpg)
EXAMPLE:
![Page 24: 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 a 2 + b 2 = c 2 a, leg.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649dd85503460f94acd798/html5/thumbnails/24.jpg)
EXAMPLE: A logo in the shape of an equilateral
triangle Find the height of the logo. Each side is 2.5 inches long.
![Page 25: 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 a 2 + b 2 = c 2 a, leg.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649dd85503460f94acd798/html5/thumbnails/25.jpg)
7.5 APPLYING THE TANGENT RATIO Trigonometric Ratio- a ratio of the
lengths of two sides of a right triangle. SOH – CAH- TOA
b
ac
B
CA
![Page 26: 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 a 2 + b 2 = c 2 a, leg.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649dd85503460f94acd798/html5/thumbnails/26.jpg)
Tangent- ratio of the length of the opposite leg to the adjacent leg of a right triangle (Round to 4 decimal places.)- “TOA”
60
7545
D
E F
![Page 27: 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 a 2 + b 2 = c 2 a, leg.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649dd85503460f94acd798/html5/thumbnails/27.jpg)
EXAMPLE Find x. Round to the nearest tenth.
17X
9
![Page 28: 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 a 2 + b 2 = c 2 a, leg.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649dd85503460f94acd798/html5/thumbnails/28.jpg)
EXAMPLE Find the height of the flagpole to the
nearest foot.
65
24ft
![Page 29: 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 a 2 + b 2 = c 2 a, leg.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649dd85503460f94acd798/html5/thumbnails/29.jpg)
EXAMPLES
![Page 30: 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 a 2 + b 2 = c 2 a, leg.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649dd85503460f94acd798/html5/thumbnails/30.jpg)
7.6 APPLY THE SINE AND COSINE RATIOS
Sine “SOH”
Cosine “CAH”
b
ac
B
CA
![Page 31: 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 a 2 + b 2 = c 2 a, leg.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649dd85503460f94acd798/html5/thumbnails/31.jpg)
EXAMPLES
![Page 32: 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 a 2 + b 2 = c 2 a, leg.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649dd85503460f94acd798/html5/thumbnails/32.jpg)
EXAMPLE Solve the right triangle formed by the
water slide.
50ft42
Z Y
X
![Page 33: 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 a 2 + b 2 = c 2 a, leg.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649dd85503460f94acd798/html5/thumbnails/33.jpg)
INVERSE TRIGONOMETRIC RATIOS Inverse tan tan-1
If tan A = x then, tan-1 x = m<A Inverse sin sin-1
If sin A = y, then sin-1 y= m<A Inverse cos cos-1
If cos A = z, then cos-1 z = m<A
b
ac
B
CA
![Page 34: 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 a 2 + b 2 = c 2 a, leg.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649dd85503460f94acd798/html5/thumbnails/34.jpg)
EXAMPLE Angle C is an acute angle in a right
triangle. Approximate the m<C is to the nearest tenth degree when:
Sin C = 0.2400
Cos C = 0.3700
![Page 35: 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 a 2 + b 2 = c 2 a, leg.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649dd85503460f94acd798/html5/thumbnails/35.jpg)
EXAMPLE Approximate the measure of angle Q to
the nearest tenth of a degree.
12
8Q S
R
![Page 36: 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 a 2 + b 2 = c 2 a, leg.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649dd85503460f94acd798/html5/thumbnails/36.jpg)
EXAMPLE A road rises 10 feet over a 200 foot
horizontal distance. Find the angle of elevation.
![Page 37: 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 a 2 + b 2 = c 2 a, leg.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649dd85503460f94acd798/html5/thumbnails/37.jpg)
EXAMPLES
![Page 38: 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 a 2 + b 2 = c 2 a, leg.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649dd85503460f94acd798/html5/thumbnails/38.jpg)
7.7 SOLVING RIGHT TRIANGLES To solve a right triangle means to find
the measures of all of the sides and angles.
You need: 2 side lengths or one side and one angle
![Page 39: 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 a 2 + b 2 = c 2 a, leg.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649dd85503460f94acd798/html5/thumbnails/39.jpg)
ANGLES OF ELEVATION AND DEPRESSION
Angle of Elevation- the angle your line of sight makes with a horizontal line while looking up
Angle of Depression- the angle your line of sight makes with a horizontal line while looking down
angl e of el evati on
angl e of depressi on
![Page 40: 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 a 2 + b 2 = c 2 a, leg.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649dd85503460f94acd798/html5/thumbnails/40.jpg)
EXAMPLE You are skiing down a mountain with an
altitude of 1200m. The angle of depression is 21o. How far do you ski down the mountain? Round to the nearest meter.
![Page 41: 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 a 2 + b 2 = c 2 a, leg.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649dd85503460f94acd798/html5/thumbnails/41.jpg)
EXAMPLE You are looking up at an airplane with an
altitude of 10,000ft. Your angle of elevation is 29o. How far is the plane from where you are standing? Round to the nearest foot.
![Page 42: 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 a 2 + b 2 = c 2 a, leg.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649dd85503460f94acd798/html5/thumbnails/42.jpg)
PERIMETER EXAMPLE
15
120 20
![Page 43: 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 a 2 + b 2 = c 2 a, leg.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649dd85503460f94acd798/html5/thumbnails/43.jpg)
UNIT CIRCLE
r
P(x,y)