Q.317. Geometry

Question 317.
Refer to the following geometric figure and prove that
(A1 + A2) - (A3 + A4) = 8A5.

Answer 317.

Refer to the following figure.

Let P be a point on GC such that GP = GE.

Draw a horizontal line through P forming a chord of the circle.
FQ is a vertical line through F meeting this chord and SR is another vertical line such that FS = EF
Thus, PRSE is a rectangle containg four equal rectangles each of area 2 * A5
=> area PRSE = 8 * A5 and is a part of A1.

Also, area above the horizontal chord throgh P and on right of CP = A4 and is also part of A1
CP = ED => area above the horizontal chord through P and on left of CP = A2
AE = SB => area between the horizontal chords on left of PE = raea on right of RS = A3 - A2 which is also part of A1.

=> A1 is made up of 8 * A5 + A4 + (A3 - A2)
=> (A1 + A2) - (A3 + A4) = 8 * A5.

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