A point is taken at random in each of the two adjacent sides of a square. Show that the average area of the triangle formed by joining them is one eighth of the area of the square. Average area of triangle formed 1/8 that of square

la1noxz

la1noxz

Answered question

2022-10-03

Average area of triangle formed 1 8 that of square.
A point is taken at random in each of the two adjacent sides of a square. Show that the average area of the triangle formed by joining them is one eighth of the area of the square.

Answer & Explanation

procjenomuj

procjenomuj

Beginner2022-10-04Added 8 answers

Step 1

In compound probability product of two different probabilities A r e a = 1 2 x y ;   p A r e a = 1 2 p x p y
Step 2
Here p x = p y = 1 2 for the full range of side length (0,1) considered
We can directly compare the area of gray triangle area to that of the square 1 2 ( 1 2 ) 2 1 2 = 1 8
seguitzla

seguitzla

Beginner2022-10-05Added 4 answers

Step 1
Inside your square, draw the quadrilateral with vertices (p,0), (0,q), (p,1), (1,q). The triangles in the corners of the square have total area 1/2. Proof: Draw the dashed lines below. Each triangle is half the area of the rectangle containing it, and those rectangles add up to area 1.

Step 2
Thus, the average area of these 4 triangles is 1/8. By the symmetry of the problem, therefore, the expected area of each of the triangles is 1/8.

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