Question
A, B and C are three points on a straight line, not necessarily equidistant with B being between A and C. Three semicircles are drawn on the same side of the line with AB, BC and AC as the diameters. BD is perpendicular to the line ABC, and D lies on the semicircle AC.


If the funny shaped diagram between the three semicircles has an area of 1000 square cms, find the length of BD.
Answers
Answer


The length of BD is 35.68 cms

There are 3 right-angled triangles - ABD, CBD and ADC.

From ABD, AB^2 + BD^2 = AD^2 ------ I
From CBD, CB^2 + BD^2 = CD^2 ------ II
From ADC, AD^2 + CD^2 = AC^2 ------ III

Adding I and II,
AB^2 + BC^2 + 2*BD^2 = AD^2 + CD^2 ------ IV

FROM III and IV
AB^2 + BC^2 + 2*BD^2 = AC^2
AB^2 + BC^2 + 2*BD^2 = (AB+CB)^2
2*BD^2 = 2*AB*CB
BD^2 = AB*CB
BD = SQRT(AB*CB)

Given that funny shaped diagram beween three semicircles has an area of 1000 square cms.
[PI/2 * (AC/2)^2] - [PI/2 * (AB/2)^2] - [PI/2 * (BC/2)^2] = 1000
PI/8 * [AC^2 - AB^2 - BC^2] = 1000
PI * [(AB+BC)^2 - AB^2 - BC^2] = 8000
PI * [2*AB*BC] = 8000
AB * BC = 4000/PI

Hence BD = SQRT(4000/PI) = 35.68 cms
where PI = 3.141592654

Hence, the length of BD is 35.68 cms.   Your Comment




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