Kate is helping plan her family's new patio

[Greg Bernhardt]

Kate is helping plan her family's new patio. It will be an irregular hexagon. They will use 1-foot-square stone tiles. Here are her notes.

1. The perimeter of the deck is outlined by ABCDEF.
2. Line AB is against the house. It is 16 feet long.
3. AF is perpendicular to AB.
4.AF and BC are parallel.
5.BC and AF are each 6 feet long.
6.Points F and C are connected by FC; FC is parallel to AB.
7.Points G and H are on FC.
8.FG and HC are each 4 feet long.
9. ED is 8 feet long. It is parallel to AB.
10. DH is perpendicular to HC. It is 4 feet long.
11. EG is perpendicular to FG. It is 4 feet long.
12. Angle BCD is 135 degrees.
13. Angle AFE is 135 degrees.

If they order 6 more tiles than they would ideally need (to allow for breakage), how many patio tiles should they order?

Assume that tiles can be cut cleanly so that any leftover pieces can be used for other parts of the deck.

 

[jamesrc]

150 tiles.

The total area of the patio is 112 ft²:

rectangle ABCF is 16*6 = 96 ft²
rectangle GHED is 8*4 = 32 ft²
and the two triangles (FGE and CHD) that make up the rest of the patio have a total area of 16 ft².

Add it together to get 144 ft².

With the 6 spare tiles, that's 150 total tiles.
(Or it isn't and I'm going to be embarrassed.)

 

[marcus]

I believe his answer was correct in the sense that
they should order 150 tiles

perhaps the statement about the patio area being 112 sq ft
is incorrect, since by his calculation it is 144
but the answer 150 tiles is right
or?

 

[jamesrc]

Originally posted by marcus

perhaps the statement about the patio area being 112 sq ft
is incorrect, since by his calculation it is 144
but the answer 150 tiles is right
or?

Ooh, right. That's just a typo. There was a bit of redundant information in the problem statement; maybe that has something to do with why I'm wrong...

 

[marcus]

Originally posted by jamesrc
Ooh, right. That's just a typo. There was a bit of redundant information in the problem statement; maybe that has something to do with why I'm wrong...

I don't understand why there is so much redundancy in Kate's notes
either. But I still think it is possible that your answer of
150 tiles is correct (that is, what Greg is looking for) and
that he has simply been busy and has not attended to this thread.

The reason I think this is possible is
1. it is christmasnewyears holidays (everyone's busy) and
2. an irregular hexagon is a PLANAR figure

since the patio is planar, there is not much room for variation of the area, so I don't see how it can be anything but 144 sq ft

BTW how did you come to accidently write 112 sq ft? was it really just a typo and you meant 144 all along (as would be consistent with the rest of your answer)?

also BTW I agree that the extreme amount of redundancy in the info is highly suspicious---just have zero idea of how to interpret it within the context of a planar figure

 

[NateTG]

Originally posted by marcus

The reason I think this is possible is
1. it is christmasnewyears holidays (everyone's busy) and
2. an irregular hexagon is a PLANAR figure

since the patio is planar, there is not much room for variation of the area, so I don't see how it can be anything but 144 sq ft

also BTW I agree that the extreme amount of redundancy in the info is highly suspicious---just have zero idea of how to interpret it within the context of a planar figure

Of course -- the trick is that you need to account for the curvature of the earth, so an additional tile is required (assuming that tiles can only be bought one at a time)

The only other thing I can come up with is if the angles are measured as external angles, then the patio is 'C' shaped and has a total area of 48 square feet, with 6 extra tiles, that's 54 tiles.

 

[marcus]

Originally posted by NateTG
Of course -- the trick is that you need to account for the curvature of the earth, so an additional tile is required (assuming that tiles can only be bought one at a time)

The only other thing I can come up with is if the angles are measured as external angles, then the patio is 'C' shaped and has a total area of 48 square feet, with 6 extra tiles, that's 54 tiles.

In the case of the 'C' shape it would seem to me that the angle BCD is either 45 degrees or (if measured as an external angle) 315 degrees. So I am still in suspense as to what the answer is supposed to be.

 

[jamesrc]

Originally posted by marcus
BTW how did you come to accidently write 112 sq ft? was it really just a typo and you meant 144 all along (as would be consistent with the rest of your answer)?

In my haste to type out an answer, I had added wrong (96+16 = 112, and I forgot to add the other 32), so I almost answered 118 tiles. When I spelled out my answer in my post, I saw my mistake and fixed it up before submitting (except for the 2nd line, obviously). That's why I added that last line to my first post; I figured that if I goofed the arithmetic once, I might have done it twice without realizing it. That and I thought I may have missed a trick, since the solution seemed so straightforward.

 

[marcus]

Originally posted by jamesrc
...That and I thought I may have missed a trick, since the solution seemed so straightforward.

I see how that could happen, and as for missing a trick, we may both be. It does seem too straightforward and there is that odd redundancy.

 

[Greg Bernhardt]

jamesrc with the point!


The area of ABCF is 96 square feet.
The area of DEGH is 32 square feet.
The area of CDH and EFG combined is 16 square feet. You will be able to complete these sections with 8 tiles each after cutting four of the tiles in half diagonally.
The total area is 144 square feet. Allowing 6 extra tiles for breakage brings the order to 150 tiles.