1. find the geometric mean between: a) 2 and 3 b) 4 and 9 c) 7 and 14 d) 15 and 60 § 13.1 a. √6...
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![Page 1: 1. Find the geometric mean between: a) 2 and 3 b) 4 and 9 c) 7 and 14 d) 15 and 60 § 13.1 a. √6 b. √36 = 6 c. √98 d. √900 = 30 The geometric mean of a](https://reader031.vdocuments.us/reader031/viewer/2022032806/56649efc5503460f94c0eee2/html5/thumbnails/1.jpg)
1. Find the geometric mean between:a) 2 and 3b) 4 and 9c) 7 and 14d) 15 and 60
§ 13.1
a. √6
b. √36 = 6
c. √98
d. √900 = 30
The geometric mean of a and b is √ab
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2. Complete each statement:a. If 4x = 7y, then x/y = ______ and y/4 = ______ .b. If 12m = 21, then 4m = ______ and m/7 = ______ .c. If 6x = 5 9, then x/5 = ______ and x/9 = ______ .d. If 15x/28y = 5a/4b, then bx = ______ and x/a = ______
a. Given 4x = 7y. Divide both sides by 4y; x/y = 7/4
Divide both sides by 28; x/7 = y/4
b. Given 12m = 21. Divide both sides by 3; 4m = 7;
Divide both sides by 84; 4/7 = 4
c. Given 6x = 5 9. Divide both sides by 30; x/5 = 9/6 = 3/2; Divide both sides by 54; x/9 =
45/54 = 5/6.d. Given 15x/28y = 5a/4b. Multiple both sides by 28by/15;
yielding bx = 7ay/3; Multiply both sides by 28y/15a;
x/a = 7y/3b
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3. Complete each statement:a. If 5/12 = 15/36, then (5 + 12)/12 = (15 + ?)/36 .b. If 7/9 = 28/36, then 7/2 = 28/(36 - ?) .c. If a/b = 6/5, then (a + b)/b = ______ and (a – b)/b =
______ .d. If (a + c)/c = 11/7, then a/c = ______ and c/a = ______ .
a. If 5/12 = 15/36, then (5 + 12)/12 = (15 + 36)/36b. If 7/9 = 28/36, then 7/2 = 28/(36 - 28) .c. If a/b = 6/5, then (a + b)/b = 11/5 and (a – b)/b = 1/5 .d. If (a + c)/c = 11/7, then a/c = 4/7 and c/a = 7/4 .
a c a b c d a b c dIf Remember then and
b d b d b d
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4. Make a table with several entries of two positive numbers and their arithmetic mean and geometric mean. Make a conjecture about a relationship between these two means. Can you prove your conjecture?
Geometric mean ≤ arithmetic mean
A B Arithmetic mean Geometric mean
1 1 1 1
2 18 10 6
3 7 5 4.5826
234 518 421 348.1551
.2 .6 .4 .3464
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4.
(a – b) 2 > 0
a 2 – 2ab + b 2 > 0
a 2 + 2ab + b 2 > 4ab
(a + b) 2 > 2√ab
(a + b) 2 /2 > √ab
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5. Research Pythagorean Triples. List five primitive triples.
Isn’t the internet wonderful!
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6. Find the length of the altitude of an equilateral triangle with side 20.
b = 10√3.
10 b 20
1 23
a b c
1 23
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7. Find the length of the altitude of an square with side 20.
c = 20√2.
a b c
1 1 2
20 20 c
1 1 2
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8. Find the longest interior dimension of a box measuring 2meters by 3 meters by 4 meters.
We will use Pythagoras twice. First to find the length of the green line and then to find the length of the magenta line.
24
3
G 2 = 2 2 + 3 2 = 13
M 2 = 4 2 + G 2 = 16 + 13 = 39 and then
M = 6.249
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10. How can you use similar triangles to find the height of the flag pole in front of the library?
h
l 1
l 2
x
2 1
x h
l l
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11. How can you use similar triangles to find the diameter of the earth?
Research Eratosthenes.
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12. Find the relationship between the areas of two similar triangles.
Area = ½ bh Area = ½ kbkh
If the sides have a ratio of k then the areas have a ratio of k 2.
b kb
hkh
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13. Prove that if a line parallel to one side of a triangle intersects the other two sides, then it cuts off a similar triangle.
Statement Reason
1. AB DE Given
3. CAB = CDE Corresponding angles
4. ∆ABC ~ ∆DEC AA
2. C = C Reflexive
A
C
B
D E
Given: AB DE.
Prove: ∆ABC ~ ∆DEC
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14. Prove SSS similarity.
Given:
Prove: ∆ABC ~ ∆DEF
Statement Reason
1. Given
3. AB/AE’ = AC/AF’ Substitution
4. A = A Reflexive
6. E’F’/BC/AE’/AB Similarity
2. AE’ = DE, AF’ = DF Construction
9. E’F’ = EF Transitive
5. ∆ABC ~ ∆AE’F’ SAS
7. E’F’ = BC· AE’/AB = BC· DE/AB Multiplication.
8. EF = BC· DE/AB Multiplication.
10. ∆AE’F’ ∆DEF SSS
D
FEB
A
C
F’E’
AB AC BC
DE DF EF
AB AC BC
DE DF EF
11. ∆ABC ~ ∆DEF (5) AND– (10)
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15. Prove that the altitude to the hypotenuse separates the triangle into two triangles which are similar to each other and to the original triangle.
Because of the right triangle and a common angle in each of the triangles it is easy to show the triangles similar by AA or AAA.
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16. Find the length of the altitude to the hypotenuse of a right triangle with legs of 15 and 20.
Use the Pythagorean Theorem to find the hypotenuse of 25. then
short side a h x
hypotenuse c b a
And you know a, b and c, so
15 hand h 12
25 20
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17. A method used by carpenters to divide a board into equal parts is to use the vertical studding of a building as parallel lines, and to place the board to be divided transversely across them. Why does this work?
Notice all of the similar triangles.
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18. In rectangle ABCD construct the diagonal AC. Construct the altitude from D to AC meeting AC at E. Prove that ∆CDE ~ ∆ABC.
Because of the right triangle and a common angle in each of the triangles it is easy to show the triangles similar by AA or AAA.
E
D C
A B
E
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19. In the figure below ∆ABC ~ ∆DAB. Prove that AB is the geometric mean between AD and BC.
Because ∆ABC ~ ∆DAB we have the following proportions -
A
H
D
C
B
H
A
D
C
B
AB BC AC
DA AB BD
And the first proportion gives us AB as the geometric mean between AD and BC>