make a square using 20 right angled triangles?

Question

3cm-one side
6cm -another side

Answer 1

Make rectangles (3x6) with pairs of the triangles.
Arrange 3 of the pairs across the first row, horizontally.
Put 1 pair vertically over the first pair and another over the 3rd pair.
Put another vertical pair above those you just placed.
Put 3 pairs horizontally across the top.
You have a square that is 18 by 18 and used 20 triangles!

See what it looks like here -
http://www.geocities.com/scientific_arts_of_monroeville/essays/square.html

OK, i got the solid version to work. Thanks to Richarda's hints.
The image is in the above link.

Answer 2

I don't know why you need 20 of them.. You only need 4.

Draw a right triangle 3cm one side, 6cm the other, Then on top of draw another triangle of the same size, but upside down. Now you will have a rectangle 3cm by 6cm..

Repeat this by drawing two more triangles above the two you have. Now you have a square that is 6 cm by 6 cm

Answer 3

Here's how you do it:
- Create six 6x3 rectangles by pairing up triangles.
- Use two of these 6x3 rectangles to make a 6x6 square
- Along each side of the 6x6 square, put another 6x3 rectangle. This gives you cross-shaped figure, basically a 12x12 square with little 3x3 squares chopped off each corner. To get to this point, you've used twelve triangles.
- Now place the eight remaining triangles along the edges of your cross. They'll fit in place quite nicely to give you a square of dimensions 6sqrt(5)*6sqrt(5). Each side of your final square shape will consist of two hypotenuses.

How did I figure it out? The first thing I did was to figure out the total area of the square. This gave me the outside side lengths. It's important to note that there can be different types of right triangles at work here... one where the hypotenuse is 6 and the legs are 3 and 3*sqrt(3), and one where the two legs are 3 and 6, and the hypotenuse is 3*sqrt(5). Which do we want?

If we use 20 triangles with legs 3 and 3*sqrt(3), the total area will be 30*sqrt(3), so each side of the square would be the square root of that in length. There's no way for us to get such a square side length using triangles with sides 6, 3, and sqrt(3).

If we use 20 triangles with legs 3 and 6, the total area will be 180, so each side of the square would be 6*sqrt(5). This is in principle doable: Each side of the square would have to consist of two hypotenuses.

Turns out this is possible to do. The pieces fall in place quite nicely as described above, once you've constructed the outside edge with eight hypotenuses.

The scphelps2 solution doesn't give you a solid square, it gives you a hollow square - the outside square has dimensions 18x18, the inside, hollowed-out part has dimensions 12x12. It is possible to make a solid square with side lengths 6*sqrt(5), as described above - and I'd have to think that a solid square is the intended solution, given that it's possible.

Neat problem!

Answer 4

Hi. Place (draw) one of the triangles then place another on top to make a 3x6 rectangle. Put two more on top of these to make a 6x6 square. These four triangles with make the point that you cannot use the 6cm side as one side adjacent to the right angle, because only four groups of four make a square and you have four left over. Therefor the 6cm side must by the hypotenuse...

WRONG!! "scphelps2' has a correct answer!! Good, I was getting confused.

Answer 5

I'll show you exactly how by drawing the individual ∆s.

Draw a set of axes: x-axis from 0 to 19, and a y-axis from 0 to 19.

Plot the following sets of 3 points and join the dots to form a ∆:

(6, 0) (6, 3) and (12, 3) is your first ∆ . .draw it.
(12, 3) (12, 6) and (18, 6) is the next
(18, 6) (15, 6) and (15, 12) the next
(15, 12) (12,12) and (12, 18) the next
(12, 18) (12, 15) and (6, 15) the next etc
(6, 15) (6, 12) and (0, 12)
(0, 12) (3, 12) and (3, 6)
(3, 6) (6, 6) and (6, 0)

Join (6, 6) to (6, 12) draw the diagonal of the rectangle.
Do the same with (12, 6) and (12, 12)
and (6, 12) and 12, 12)
and (6, 9) and (12, 9)
and (6, 6) and (12, 6)

then draw the last missing diagonal - you should now have your square and the 20 ∆s

Answer 6

the triangles will be;
3,6, sqrt45
total area =20*3*6
=180 cm^2

with eight hypotenuse facing
outwards,we have;
(2*sqrt45)^2=4*45
=180 cm^2,it is therefore
possible

1)fit the triangles in such
a way that the eight outside
triangles form a square
{hypotenuse facing outward}
2*sqrt45x2*sqrt45

2)fill the other twelve triangles
in like a jigsaw

you have a180 cm^2 square!

that's logic

i hope that this helps

Answer 7

Perimeter = (x) + (3x) + (3x + 2) = 37 (given) ie 7x + 2 = 37 ie x = 5 for this reason facets are 5, 15 and 17 instruments. enable the backside = x ( to be calculated ) As according to the given subject section = x + sixteen top = 6 Now component of the triangle = base x a million/2 of the top. = x . (6/2) = 3x yet section is likewise equivalent to ( x + sixteen ) for this reason 3x = x + sixteen ie 2x = sixteen ie x = 8 instruments for this reason length of the backside x = 8 instruments. Perimeter of the triangle = 3 x sixteen = 40 8 m Perimeter of the sq. = 4.x the place x is the dimensions of the two facet of the sq.. for this reason 4x = 40 8 m ie x = 12m ............... answer enable the width of the rectangle = x ft for this reason its length = 2x ft. that's given the fringe = 60 ft ie 2 ( 2x + x ) = 60 ft ie 6x= 60 ie x = 10 ft Now component of the rectangle = length x breadth = x.2x = 2.sq. of x = 2 ( Sq. of 10 ) = 2 hundred squareft for this reason the facet of the isosceles triangle could have any finished selection from 3 to seventy 5. the unique Dimensions are 9.6" x 38.4" ...... answer If x is the degree of all of the equivalent angles Then the third perspective = 2x - 36 yet sum of those angles = one hundred eighty ie 2x + 2x - 26 = one hundred eighty ie x = fifty 4 degree hence angles are fifty 4 deg, fifty 4 deg, and seventy two deg .................... answer

Answer 8

A square would have equal sides, that would end up being a rectangle.

Answer 9

a square has equal sides thats a rectangle

Answer 10

Well Look At This Then... http://uk.pg.photos.yahoo.com/ph/aepacino/detail?.dir=/992ascd&.dnm=93d8scd.jpg