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All vertices of square are joined with a common random point inside square. Find ratio of sum of areas of triangles formed either sides of that point (Upper + Lower or Left + Right) to area of square.
Give this question a try then check your answer with solution.
(Shortcut Image Solution with Calculations is gives below in this post)
Solution 1:
Step 1:
Draw a perpendicular line on upper and lower side passing through that point.
Let side of square be x.
Step 2:
Let height of upper triangle be x,
⇒ Height of lower triangle be (x - a)
⇒ Area of upper triangle be ax / 2
⇒ Area of lower triangle be (x - a)x / 2
⇒ Sum of Blue area be (a + (x - a))x / 2 = x² / 2
Also area of square = x²
Step 3:
Ratio of areas = x² / 2 : x² = 1 : 2
Image solution:
Solution 2:
Step 1:
Draw a perpendicular line on upper and lower side passing through that point.
Draw a perpendicular line on left and right side passing through that point.
Step 2:
Now you can observe 4 pairs of similar triangles.
Let their areas are a1, a2, a3, a4.
Blue area = a1 + a2 + a3 + a4 = A
Square area = 2(a1 + a2 + a3 + a4) = 2A
Step 3:
Ratio of areas = A : 2A = 1 : 2
Image solution:
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