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Put in simplest radical form. (No decimals)
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Geometry Chapter 8 - Right Triangles
5.7 Notes on Right ∆’s
Given: any 3 sides of a ∆ Prove: the ∆ is acute, obtuse, or right (hint: use the converse of Pythagorean Theorem)
If the (longest side)2 > (side)2 + (side)2 then the ∆ is an ___________ ∆
If the (longest side)2 < (side)2 + (side)2 then the ∆ is an ___________∆.
If the (longest side)2 = (side)2 + (side)2 then the ∆ is a ___________ ∆.
Pythagorean Theorem:
Pythagorean Triple:
For problems 13.--, determine if it is possible to form a triangle with the given sides. If so, tell whether the
triangle is right, acute, or obtuse.
1. 3.5, 8, 6
6. 6, 8, 10
9.
3, 4 , 5
12. 10, 11, 13
2. 3, 5, 6
3. 5 2 , 5, 5
7. 4, 9, 11
10. 5, 5, 12
13. 9, 40, 41
4. 2 3 , 7 2 , 15
8. 1, 2, 3
11. 7, 8, 16
Ex. Pythagorean Theorem - Solve for x. (Simplest radical form)
1. 32 + 42 = x2
2. x2 + 42 = 52
4. x2 + 32 = 42
5. 42 + 72 = x2
2
7. 12 + x2 = 32
8. x2 + 52 = 5 2 
5. 8 3 , 8, 16
3. 52 + x2 = 132
6. x2 + 52 = 102
9. x2 + 7 3 2  2x 2
PYTHAGOREAN THEOREM
Show all necessary work. Make a sketch if needed.
1. A rectangle has length 2.4 and width 1.8. Find the length of a diagonal.
2. A rectangle has a diagonal of 2 and length of
3.
4.
5.
6.
3 . Find its width.
Find the length of a diagonal of a square with perimeter 16 m.
Find the length of a side of a square with a diagonal of length 12 inches.
The diagonals of a rhombus have lengths 16 and 30 cm. Find the perimeter of the rhombus.
The perimeter of a rhombus is 40 cm, and one diagonal is 12 cm long. How long is the other diagonal?
7. Find the third side of a right triangle if the hypotenuse is 14 km and one side is 9 km.
8. A rectangle is 6 ft long and 11 ft wide. What is the length of the diagonal of the rectangle?
9. A pole is 10 ft high. A wire is attached to the top of the pole and fastened to an anchor in the ground. The
anchor is 5 ft from the bottom of the pole. What is the length of the wire?
10. A tower casts a shadow 40 m long. The distance from the top of the tower to the end of the shadow is 50 m.
How high is the tower?
11. Marie and Kevin hiked 3 miles east and then 6 miles north. How far were they from their starting point?
Ex. Find the value of x.
1.
2.
3.
2
x
11
14.
13
15.
x
13
15
12
15
10
21
5
16.
17.
18.
x
8
11
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5.8 Special Right ∆’s
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Find the value of the variables.
Use the given length to find each of the remaining two lengths.
1. g = 6 ________________
2. h = 3 2 ______________
3. f = 9
_________________
4. f = 7 _______________
5. g = 12 _________________
6. h = 5 _______________
7. t = 4 __________________
8. u = 2 3 ______________
9. u = 10 _________________
11. t = 5 ________________
10. s = 4 3 _______________
12. s = 8 _______________
45
g
h
f
Use the diagram to find the remaining lengths.
13. a = 2, b = _________, c = ________, d= _________, e = _________
14. a = ________, b = _________, c = ________, d= _________, e = 4
15. a = ________, b = _________, c = 10 , d= _________, e = _________
Use special right ∆’s to solve:
1. Find the length of a diagonal of a square with sides 10 inches.
2. Find the perimeter of a square whose diagonal is 4 cm.
3. One side of an equilateral ∆ has length 6 cm. Find the length of the altitude.
4. Find the perimeter of an equilateral ∆ if the altitude has length 9 cm.
5. Find the length of a side of an equiangular ∆ whose altitude is 12.
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8.1 Similarity in Right Triangles
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8.2-2.3
Trigonometric Ratio
Abbreviation
Definition
Sine of P
Sin P
opposite leg to P
O

hypotenuse
H
Cosine of P
Cos P
adjacent leg to P
A

hypotenuse
H
Tangent of P
Tan P
opposite leg to P
O

adjacent leg to P
A
Ex 1 : Given the following right triangle, write the trig ratios in fraction and decimal form.
Sin P =
Sin Q =
Cos P =
Cos Q =
Tan P =
Tan Q =
Ex 2: Find the following trig values using the calculator.
(Make sure you calculator is in degree mode and then round to four decimal places)
Sin 38° =
Cos 38° =
Tan 38° =
0.615
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Sin 52° =
Cos 52° =
Tan 52° =
In Examples 3-11, find the value of the missing variable(s).
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9
Angle of Elevation:
Angle of Depreession:
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8.4 Make a sketch with a right ∆, label, show trig equation.
1. A tree casts a shadow 21m long. The angle of elevation of the sun is 51. What is the height of the tree?
2. A helicopter is hovering over a landing pad 100 m from where you are standing. The helicopter’s angle of
elevation with the ground is 12. What is the altitude of the helicopter?
3. You are flying a kite and have let out 80 m of string. The kite’s angle of elevation with the ground is 40. If
the string is stretched straight, how high is the kite above the ground?
4. A 15 m pole is leaning against a wall. The foot of the pole is 10 m from the wall. Find the angle the pole
makes with the ground.
5. A guy wire reaches from the top of a 120m television transmitter tower to the ground. The wire makes a 63
angle with the ground. Find the length of the guy wire.
6. An airplane climbs at an angle of 18 with the ground. Find the ground distance the plane travels as it moves
2500 m through the air.
7. A lighthouse operator at a point 25 m above sea level sights a sailboat. The angle of depression of the sighting
is 10. How far is the boat from the base of the lighthouse?
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8. A 20 ft ladder leans against a wall so that the base of the ladder is 8 ft from the base of the building. What
angle does the ladder make with the ground?
9. At a point on the ground 50 ft from the foot of a tree, the angle of elevation to the top of the tree is 53.
Find the height of the tree.
10. From the top of a lighthouse 210 ft high, the angle of depression of a boat is 27. Find the distance from the
boat to the foot of the lighthouse. The lighthouse was built at sea level.
8.5 The Law of Sines and Law of Cosines
Example 1 - Solve the triangle. Round to the nearest tenth.
a.
b.
c.
Use the Law of Sines to find each measure. Round lengths to the nearest tenth and
angle measures to the nearest degree.
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10.
11.
BC ________
13.
12.
DE ________
14.
mJ ________
GH ________
15.
mR ________
mT ________
Use the Law of Cosines to find each measure. Round lengths to the nearest tenth and
angle measures to the nearest degree.
16.
17.
YZ ________
19.
18.
BD ________
20.
mI ________
EF ________
21.
mM ________
mS ________
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