# Cut Right Triangle Problem

So, the other day, i was programming something and got a geometry doubt. I apologize for the bad drawing. But anyway, this is a 90-45-45 triangle with legs of 1 unit. It is cut from the 45° point to the exact middle of each of the legs. The red lines are perpendicular to the legs and extends till the meeting point of both the cuts. What is the length of the red lines (only one of them as they form a square) ?

(diagram not to scale)

Note by Akshaj Gopalakrishnan
11 months ago

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You can also use similar triangles. Let the length of the red lines be $x$. Consider the right triangle with the vertical red line as it's vertical leg. This right triangle has height $x$ and base length $1 - x$, and is similar to the right triangle with height $1/2$ and base length $1$. By similarity we then have that

$\dfrac{x}{1 - x} = \dfrac{1/2}{1} \Longrightarrow 2x = 1- x \Longrightarrow 3x = 1 \Longrightarrow x = \dfrac{1}{3}$.

- 11 months ago

Simply use co-ordinate geometry.....

- 11 months ago

Could you please explain that? I am really interested.

- 11 months ago

Here is an outline of my method.....

Consider the origin to be the right angled vertex. The remaining two points are located on $\left(1,0\right)\ \&\ \left(0,1\right)$
Hence, the midpoints will be located at $\left(\frac{1}{2},0\right)\ \&\ \left(0,\frac{1}{2}\right)$
Now, simply write the equations of the lines joining the points and you'll arrive at the answer......

- 11 months ago

Thanks!

- 11 months ago

You are welcome!!

- 11 months ago

I am getting the lenghts to be $\displaystyle \frac{1}{3}$

- 11 months ago

Bhai muje apni I'd de sakta hai plzz..

- 1 month ago

kyon kya kaam hai?? also, do I know you??

- 1 month ago

Na, muje 1 course ke aaghe ke chapter dekhne the or ye free ni hai tab.

- 1 month ago

main premium nhi hun, aur main course nhi dekhta…..

- 1 month ago

OK thanks BTW :D

- 1 month ago