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!

AnonymousConfusedGuy
Mar 29, 2018

#2**+1 **

1. Let the largest angle = a_{5}

Let the smallest angle = a_{1}

So

a_{5} = a1 + 4d

a_{5} - a_{1} = 4d

But a_{5} - a_{1} = 44

So d = 11

So the sum of the angles = 540

a_{1} +( a_{1}+ d) + (a_{1} + 2d) + (a_{1} + 3d) + (a_{1} + 4d) = 540

5a_{1} + 10d = 540

5a_{1} + 10(11) = 540

5a_{1} + 110 = 540

5a_{1} = 430 divide both sides by 5

a_{1} = 86° = the smallest angle

CPhill
Mar 29, 2018

#3**+1 **

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°

CPhill
Mar 29, 2018

#4**+1 **

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....!!!

CPhill
Mar 29, 2018

#5**0 **

Thanks so much CPhill, turn's out #3 was 18 but that might have been a glitch or a typo.

AnonymousConfusedGuy
Mar 29, 2018