Egyptian Model Building

Geometry Level 2

Acenith has been tasked by the Pharaoh with constructing a model of the next pyramid to be built. He wants his scale model to be perfect, but the usual trick of using ropes with length 3, 4, and 5 to make a right angle won't work (as he doesn't have a rope).

However, he does have boards of the following lengths: 2, 3, 5, 9, 10, 12, 15, and 20.

What is the least number of boards he could use (without cutting) to lay out a right angle and ensure that his model looks right for Pharaoh's inspection?

5 3 It isn't possible without cutting. 4

8 solutions

Test User
Feb 14, 2014

Because primitive Pythagorean Triples can be multiplied by a constant factor to get other Pythagorean Triples, we can note that $3, 4, 5$ leads to another Pythagorean Triple $9, 12, 15$ when multiplied by a factor of $3$ . This can be verified numerically: $9^2 + 12^2 = 15^2$

Cool. I saw 5, 12, 13, and got 4 as my answer (13 is 10 + 3). Great problem!

Finn Hulse - 7 years, 3 months ago

he's making a scale model so it has to be a multiple of 3-4-5

Nathan Ramesh - 7 years, 3 months ago

Did. Not. See. That. :)

Finn Hulse - 7 years, 3 months ago

right

Aditya Sher - 7 years, 3 months ago

Whilst I used Pythagorean triples to identify the solution, I'm actually ashamed to admit I was unaware of the fact that any constant multiple of a Pythagorean trple is itself a Pythagorean triple. However, I now know for the future. Thank you. :)

Alex Panebianco - 7 years, 3 months ago

Saikrishnareddy Mogilla - 7 years, 3 months ago

Yea, I used this property to solve too.

Kou$htav Chakrabarty - 7 years, 3 months ago

I tried the trial-and-error method

Rohit Nair - 7 years, 3 months ago

yeah..got it

DOLLY KRISHNA - 7 years, 2 months ago

Avani Chopra
Feb 15, 2014

Use boards of length 12 , 9 and 15 to make a right angle as (12X12) + (9X9)= 144 + 81 = 225 = (15X15)

Utkarsh Yadav
Feb 24, 2014

Even if he does not have any rope . he can have a look for boards which form a pythagorean triplet. In the given boards 9,12 and 15 form a pythagorean triplet as 9^2+12^2=15^2 . Or he can multiply the pythagorean triplet (3,4,5) by a constant factor to get another pythagorean triplet( here the constant factor is 3) to get another pythagorean triplet. Therefore the minimum number of boards required are 3.

Virok Sharma
Feb 17, 2014

The problem may be easily solved by just taking into notice the fact that 81 + 144 = 225.

Sameer Pai
Feb 16, 2014

He wants a scale model for 3,4, and 5. If the scale is x2, there is no board length 3x2=6. If the scale is x3, there is 9,12,and 15, and therefore the process can be done with 3 boards.

Mahbubur Rahman
Feb 16, 2014

I have to seek at least 3 boards having the ratio of length 5:4:3.

Fortunately, 15,12,9 meet this ratio.

So, it is possible with only * 3 boards. *

Mietantei Conan
Feb 16, 2014

See this question is very straight forward; just find one triplet of board which satisfy the
$PythagorusTheorem$ . I found $(9,12,15)$ .

Not just the pythagorus theorem, but the correct ratio too.

Sameer Pai - 7 years, 3 months ago

Aritri Chatterjee
Feb 25, 2014

15 12 9 is another triplet of this sort.

Egyptian Model Building

8 solutions

0 pending reports