Question 5 - Figure it out - Page 160 - Chapter 7 Class 8 - Area (Ganita Prakash II)

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Question 5 Give a method to obtain a quadrilateral whose area is half that of a given quadrilateral. We follow these steps Draw a diagonal line connecting two opposite corners. (Now you have two triangles). Find the exact midpoint (center dot) of that diagonal line. Draw a line from the other two corners to that center dot. Let’s look at it step-by-step Halving Algorithm Back Next Step Step 0: The Original Shape Here is a standard 4 -sided shape (quadrilateral ABCD). Drag the points to make it as weird and uneven as you want!Halving Algorithm Back Next Step Step 1: Draw a Diagonal We draw a line from corner A to corner C . This splits the big shape into two separate triangles: and .Halving Algorithm Back Next Step Step 2: Connect the Midpoint We find the exact center of the diagonal line (Point M). Then we draw lines to it from B and D to form the new green shape (ABMD). Why is Green (ABMD) exactly half? Top Triangle: Line BM splits the top triangle ( ) into two. Because M is the midpoint, they have the same base length ( MC). They share the same height. Therefore, the green piece equals the empty piece: Area( . Bottom Triangle: The same rule applies below! Line DM splits in half, so the bottom green piece equals the bottom empty piece: . Add If we add the left pieces together and the right pieces them together: upl Green Shape (ABMD) = Empty Shape (CBMD). Since the two halves are perfectly equal and make up the whole shape, they must each be exactly of the total area!The Geometric Proof Drawing the diagonal line (𝐴𝐶) splits your random 4-sided shape into a "Top Triangle" ( △𝐴𝐵𝐶 ) and a "Bottom Triangle" ( △𝐴𝐷𝐶 ). Look at the top triangle ( △𝐀𝐁𝐂 ). Point M is exactly in the middle of its base, so base AM = base MC. Since both sides share the same height, the left green piece is perfectly equal to the right empty piece. Area(△ABM)=Area (△CBM) Look at the bottom triangle ( 𝚫𝐀𝐃𝐂). The exact same rule applies. 𝑀 is in the middle, so the left green piece is perfectly equal to the right empty piece. Area (△ADM)=Area(△CDM) The Green Shape (ABMD) is made of: 𝜟𝑨𝑩𝑴+𝜟 ADM The Empty Shape (CBMD) is made of: 𝜟𝑪𝑩𝑴+𝚫 CDM Because the pieces that make them up are identical in area, the two final shapes must be identical in area! 𝐀𝐫𝐞𝐚(𝐀𝐁𝐌𝐃)=𝐀𝐫𝐞𝐚(𝐂𝐁𝐌𝐃)=𝟏/𝟐 𝐀𝐫𝐞𝐚 (𝐀𝐁𝐂𝐃)