3, 4, 5 Triangle
Need a Right Angle (90°) Fast ... ?
Make a 3,4,5 Triangle ! Connect three lines:
And you will have a right angle (90°) |
Other Lengths
You can use other lengths by multiplying each side by 2. Or by 10. Or any multiple.
Drawing It
Let us say you need to mark a right angle coming from a point on a wall.
You decide to use 300, 400 and 500 cm lines.
Draw a 300 line along the wall | Draw an arc 400 away from the start of the 300 line | Draw an arc 500 away from the end of the 300 line | Connect from the start of the 300 line to where the arcs cross | And you have your "3,4,5" triangle with its right angle |
The Mathematics Behind It
The Pythagoras Theorem says:
In a right-angled triangle, the square of a (a^{2}) plus the square of b (b^{2}) is equal to the square of c (c^{2}): a^{2} + b^{2} = c^{2} |
Let's check if it does work: 3^{2} + 4^{2} = 5^{2} Calculating this becomes: 9 + 16 = 25 Yes, it works ! |
Other Combinations
Yes, there are other combinations that work (not just by multiplying). Here are two others:
5,12,13 triangle | 9,40,41 Triangle | |
5^{2} + 12^{2} = 13^{2} | 9^{2} + 40^{2} = 41^{2} | |
25 + 144 = 169 | (try it yourself) |
And there are infinitely many more.... read Pythagorean Triples for more information.
But Wait ... There is More!
The 3,4,5 Triangle has a simple beauty. At its heart is a circle:
The circle is a Unit Circle: it has a radius of exactly 1 and an area of π
And using that circle we can cut our triangle into six little triangles like this:
Each small triangle is also a right triangle!
And notice that 1+2=3, 1+3=4 and 2+3=5, making the 3,4,5 triangle. So simple and neat.
Is the Radius Really 1?
Imagine we don't know the radius, call it "r" and draw in what we do know: