1. **Hexagon Help**

I am stuck with this question, can anyone help?

Hexagon ABCDEF has the following properties:
i. diagonals AC, CE and EA are all the same length
ii. angles ABC and CDE are both 90 degrees
iii. all the sides of the hexagon have lengths which are different integers

a) What is the minimum perimeter of ABCDEF if AC = √85 (sqr root of 85)
b) What is the smallest length of AC for which ABCDEF has all these properties? What is the minimum perimeter in this case?

Any help would be much appreciated

thanks

2. Originally Posted by Chuck_3000
I am stuck with this question, can anyone help?

Hexagon ABCDEF has the following properties:
i. diagonals AC, CE and EA are all the same length
ii. angles ABC and CDE are both 90 degrees
iii. all the sides of the hexagon have lengths which are different integers

a) What is the minimum perimeter of ABCDEF if AC = √85 (sqr root of 85)
b) What is the smallest length of AC for which ABCDEF has all these properties? What is the minimum perimeter in this case?

Any help would be much appreciated

thanks
First the diagram: see attachment.

Now Part (a):

Now $\displaystyle \triangle\ ABC$ and $\displaystyle \triangle\ CDE$ are right triangles with hypotenuses of length
$\displaystyle \sqrt{85}$, so for $\displaystyle \triangle ABC$:

$\displaystyle AB^2+BC^2=85$

with $\displaystyle AB$ and $\displaystyle BC$ integers, and trail and error shows that the pair
of lengths $\displaystyle (AB,BC)$ is one of $\displaystyle (2,9),\ (6,7),\ (9,2),\ (7,6)$.

The same argument applies to $\displaystyle \triangle CDE$.

So for part (a) we have the minimum of $\displaystyle AB+BC+CD+DE=24$
(each pairs $\displaystyle (AB,BC),\ (CD,DE)$ is one of $\displaystyle (2,9),\ (6,7),\ (9,2),\ (7,6)$
but with none of the sides equal, so these sides is one of $\displaystyle 2,6,7,9$ in some order, without repetition)

Also $\displaystyle EF+FA$ cannot be $\displaystyle 10$ or less (trial and error shows this can't be
$\displaystyle 10$, and it must be greater than $\displaystyle 9$ as it must be greater than $\displaystyle \sqrt{85}$).
But $\displaystyle EF+FA$ can be $\displaystyle 11$.

So the minimum perimeter is $\displaystyle 24+11=35$.

RonL

3. Originally Posted by Chuck_3000
I am stuck with this question, can anyone help?

Hexagon ABCDEF has the following properties:
i. diagonals AC, CE and EA are all the same length
ii. angles ABC and CDE are both 90 degrees
iii. all the sides of the hexagon have lengths which are different integers

a) What is the minimum perimeter of ABCDEF if AC = √85 (sqr root of 85)
b) What is the smallest length of AC for which ABCDEF has all these properties? What is the minimum perimeter in this case?

Any help would be much appreciated

thanks
Hello,

to a)
You have to do with right triangles. The squares of the legs must be 85, that means, you have to split 85 into 2 squares. Thus:
AB = 2
BC = 9

DC = 6
ED = 7

Now you are looking for integers, which are not present in the list of legs above(1, 3, 4, 5, 8). The sum of these integers must exceed √(85) approximately 9.22. Therefore you have as possible combination:

EF = 3, AF = 8
EF = 4, AF = 8
EF = 5, AF = 8

I'm awfully sorry, but I can't offer you any help for part b).

Greetings

EB

4. thank you so much guys!

CaptainBlack, how do you get those images on there?

5. Originally Posted by Chuck_3000
I am stuck with this question, can anyone help?

Hexagon ABCDEF has the following properties:
i. diagonals AC, CE and EA are all the same length
ii. angles ABC and CDE are both 90 degrees
iii. all the sides of the hexagon have lengths which are different integers

a) What is the minimum perimeter of ABCDEF if AC = √85 (sqr root of 85)
b) What is the smallest length of AC for which ABCDEF has all these properties? What is the minimum perimeter in this case?

Any help would be much appreciated

thanks
part (b)

Just an idea:

The smallest length for AC is the square root of the smallest number
that can be written as the sum of two squares in two different ways
with no common square among the four.

RonL

6. Originally Posted by Chuck_3000
thank you so much guys!

CaptainBlack, how do you get those images on there?
I use PowerPoint as a drawing package, then capture the image as
a bitmap into Paint Shop Pro (or any other photo editor) then
resize and save as a JPEG.

Then when composing a post I upload the image from the manage
attachments dialog.

RonL

7. Originally Posted by Chuck_3000
I am stuck with this question, can anyone help?

Hexagon ABCDEF has the following properties:
i. diagonals AC, CE and EA are all the same length
ii. angles ABC and CDE are both 90 degrees
iii. all the sides of the hexagon have lengths which are different integers

a) What is the minimum perimeter of ABCDEF if AC = √85 (sqr root of 85)
b) What is the smallest length of AC for which ABCDEF has all these properties? What is the minimum perimeter in this case?...
Hello,

$\displaystyle \sqrt{65}=1^2 + 8^2 = 4^2 + 7^2$

Greetings

EB

8. An almost entirely unrelated question.

Doesn't "hexagon" imply equal sides? So wouldn't the question be asking about a "hexalateral" or some such?

Just curious.

-Dan

9. Originally Posted by topsquark
An almost entirely unrelated question.

Doesn't "hexagon" imply equal sides? So wouldn't the question be asking about a "hexalateral" or some such?

Just curious.

-Dan
No hexagon=six sided planar figure, with all sides equal we are talking about
a regular hexagon.

RonL

10. Originally Posted by CaptainBlack
No hexagon=six sided planar figure, with all sides equal we are talking about
a regular hexagon.

RonL
Now I just feel silly. I knew that. (Ahem!)

-Dan

11. Originally Posted by CaptainBlack
No hexagon=six sided planar figure, with all sides equal we are talking about
a regular hexagon.

RonL

I can make that a large hexagon for an extra 50 cents,
and would you like fries with that?

RonL

12. Originally Posted by CaptainBlack
I can make that a large hexagon for an extra 50 cents,
and would you like fries with that?

RonL
SuperSize me!

Wait, don't. I'm big enough already!

-Dan