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# Three unrelated questions

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1:  The interior angle measures of a pentagon form an arithmetic progression. The difference between the largest and smallest angle measures is 44 degrees. Find the measure of the smallest angle, in degrees.

2:  In the diagram, square ABCD has sides of length 4, and triangle ABE is equilateral. Line segments BE and AC intersect at P. Point Q is on BC so that PQ is perpendicular to BC and PQ=x. Determine the measure of angle BPC.

3:  Fido's leash is tied to a stake at the center of his yard, which is in the shape of a regular hexagon. His leash is exactly long enough to reach the midpoint of each side of his yard. If the fraction of the area of Fido's yard that he is able to reach while on his leash is expressed in simplest radical form as [(sqrt{a})/b] * pi, what is the value of the product ab?

(It's not 2sqrt3)

Thanks so much!

Mar 29, 2018
edited by AnonymousConfusedGuy  Mar 29, 2018

### 5+0 Answers

#1
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I got #1, the answer is 86 degrees :)

Mar 29, 2018
#2
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1.  Let the largest angle   = a5

Let the smallest angle   =  a1

So

a5 = a1  + 4d

a5 - a1  =  4d

But  a5 - a1  = 44

So  d  = 11

So the sum of the angles = 540

a1 +( a1+ d)  + (a1 + 2d) + (a1 + 3d) + (a1 + 4d)   =  540

5a1 + 10d  =  540

5a1 + 10(11) = 540

5a1  + 110   = 540

5a1  = 430           divide both sides by 5

a1  = 86°  =   the smallest angle   Mar 29, 2018
edited by CPhill  Mar 29, 2018
#3
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2.

Since ABE is equilateral....then angle ABE  = 60°

And since ABCD  is a square, then angle EBC =  90 - 60  = 30°

And since AC is a diagonal of the square, then  angle ACB  = 45°

Then, in triangle BPC,   angle BPC  must be   180  - 30 - 45  =  180  - 75  = 105°   Mar 29, 2018
#4
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3.

Let the side of the hexagon  = S

Then....the area  of the hexagon =  6(1/2)S^2*(√3/ 2)  = 3√3/2 S^2  units^2

Now ...to determine the length of Fido's leash, L, we have

(1/2)S             L

_____   =      ______

sin (30)         sin (60)

(√3/2)S  =  L

So....the area  that Fido can cover is     pi  (√3/2S)^2  =  (3/4)pi*S^2 units^2

So  the fraction of the yard that Fido can cover is

(3/4)pi*S^2               (3/4)  pi                    6 pi               pi          √3

___________  =    ___________  =    ______  =    _____  =   ___  pi

3√3/2 S^2                  3√3/2                  12√3               2√3          6

So   a  = 3   and b  =  6

And the product ab  is 6*3   =  18

EDIT TO CORRECT AN ERROR....!!!   Mar 29, 2018
edited by CPhill  Mar 29, 2018
#5
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Thanks so much CPhill, turn's out #3 was 18 but that might have been a glitch or a typo.

Mar 29, 2018