MHB Find the ratios of the areas of the four regions

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The discussion focuses on finding the area ratios of four regions formed in square ABCD by point M, the midpoint of side CD. It highlights that triangles MPC and APB are similar, which is crucial for determining the area ratios. Participants suggest using properties of similar triangles to derive the necessary ratios. The conversation emphasizes the importance of justification in the calculations to ensure accuracy. Overall, the goal is to clearly establish the area ratios of ∆MPC, ∆BPC, ∆APB, and quadrilateral APMD.
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In square ABCD point M is the midpoint of side CD. Find the ratios of the areas of the four regions (∆MPC, ∆BPC, ∆APB, and quadrilateral APMD) that are formed. Justify your result.View attachment 8047
 

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Re: Having a difficult time figuring out how to solve this

Hello, and welcome to MHB! (Wave)

I think my first step would be to explain how $\triangle MPC$ and $\triangle APB$ are similar, from which we can determine the ratio of their areas...
 

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Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...
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