Finding the height
Geometry
Level
2
In triangle ABC, angle B is perpendicular. Let D be a point on side AC such that angle D is right. If the sum of AB and BC is 35 and the sum of BD and AC is 37, find the height, BD.
The answer is 12.
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(AB + BC)^2 = AB^2 + 2AB(BC) + BC^2
(35)^2 = AB^2 + 2AB(BC) +BC^2
AB^2 + BC^2 = AC^2
therefore,
1225 = AC^2 +2AB(BC)
transpose AC^2 and divide both sides by 4.
(1225 - AC^2)/4 =( AB(BC))/2
(AB(BC))/2 is the area of the triangle......therefore,
(AC(BD))/2 = (1225 - AC^2)/4
BD = 37 - AC
therefore,
(AC(37 - AC))/2 = (1225 - AC^2)/4
2*(37AC - AC^2 ) = 1225 - AC^2
74AC - 2AC^2 = 1225 - AC^2
AC^2 - 74AC + 1225 = 0
(AC - 25) (AC - 49) = 0
Since AC + BD = 37 only.....then AC = 25
and.........BD = 37 - 25 = 12