1. ## Probability Problem

Hello, am new to this website and am sorry if I posted this post in the wrong topic. I would be very grateful if you could help me out with this. Thank you.

Well...here is my problem:

M and N are the midpoints of adjacent sides of the square WXYZ. A point is selected
at random in the square. Calculate the probability that is lies in triangle MYN.

2. What is the ratio of the area of the triangle to the area of the rectangle?

3. hehe...that's the issue, i don't have any other information except those given above...

4. Originally Posted by Yudhishm
hehe...that's the issue, i don't have any other information except those given above...
The answer is $\displaystyle \frac{1}{8}$. WHY?

5. yes and why?

6. That is for you to do.

7. yeah...but i tried to figure it out SEVERAL TIMES! and i still can't get it...How did you do it?

8. Alright for all those who did not know, here is the answer, i just figured it out

P(the point lies in triangle MYN) = Area of triangle MYN / Area of the square WXYZ

Let the area of the triangle be: A
and let the area of the square be: B
-------------------------------------------------------------
We all know that area of a triangle is 1/2 * base * height

Therefore, A = 1/2 * 1/2 * 1/2
= 1/8
--------------------------
Now B,

B = Length * Height

Since M and N are the midpoint of ZNY and XMY:

B = (1/2 + 1/2) * (1/2 + 1/2)
= 1

Now: A/B = 1/8 / 1 = 1/8

Thank you, Plato

9. Originally Posted by Yudhishm
Alright for all those who did not know, here is the answer, i just figured it out

P(the point lies in triangle MYN) = Area of triangle MYN / Area of the square WXYZ

Let the area of the triangle be: A
and let the area of the square be: B
-------------------------------------------------------------
We all know that area of a triangle is 1/2 * base * height

Therefore, A = 1/2 * 1/2 * 1/2
= 1/8
--------------------------
Now B,

B = Length * Height

Since M and N are the midpoint of ZNY and XMY:

B = (1/2 + 1/2) * (1/2 + 1/2)
= 1

Now: A/B = 1/8 / 1 = 1/8

Thank you, Plato
You implicitly assumed that the sides of the square have length 1, which is fine here. More generally you could choose a variable name such as s to represent the side length, then show that it cancels out when taking the ratio.

Much easier way is to look at symmetry and see pretty much instantly that the answer must be 1/8. Imagine dividing the square WXYZ into four smaller squares. Then triangle MYN obviously has the area of half of one of those smaller squares. And you are done.

10. I just tried as far as possible to get into details. Anyway, thank you a lot

11. Another solution...

Look at the picture...

12. oh great...thank you a lot (happy)