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# Geometry

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+1141

Can anyone give a hint here?

In square ABCD, point M is the midpoint of CD.  Find the ratios of the areas of the four reigons that are formed.

Dec 31, 2023

#1
+129270
+1

For convenience, let the sides of the square =  2

Let

B = (0,0)

C= (2,0)

D =(2,2)

A =(0,2)

M = (2,1)

Line through AC has the slope  -1

The equation of this  line  is  y= -1 (x -2)  =  -x + 2

Line through BM  has a sloe of (1/2)

Equation of  this  line is   y = (1/2)x

x intersection of these two lines

-x + 2 = (1/2)x

2 = (3/2)x

x = 4/3

y = (1/2)(4/3) = (4/6)  = (2/3)

So the coordinates  of P   =( 4/3 , 2/3)

Area of BPC = (1/2)(BC)(2/30 =  (1/2)(2)(2/3)  =  2/3

Area of APB =  (1/2) (AB)(4/3) =  (1/2)(2)(4/3) =  4/3

Area of MPC =  (1/2)(MC) (2 -4/3)  =( 1/2) (1) (2 - 4/3)  =  (1/2)(2/3)  =  2/6  = 1/3

Area of APMD =  area of square less the other areas =   4 - 2/3 - 4/3  - 1/3 =  4 - 7/3  = 12/3 - 7/3 = 5/3

The problem wants to  find the ratios  of the areas to each other ???....if so

MPC : BPC : APB : APMD =   1 : 2 : 4 : 5

Jan 1, 2024