Asked by Sheenybeany
In quadrilateral ABCD, we have AB = BC = CD = DA, AC = 14, and BD = 48. Find the perimeter of ABCD.
Answers
Answered by
MathMate
A quadrilateral with all four sides equal is a rhombus, whose area is <i>half</i> the product of the diagonals AC and BD.
Answered by
Sheenybeany
Yeah, but we're not trying to find the area?
Answered by
MathMate
Sorry.
In that case, we make use of the fact that rhombuses have diagonals which intersect at right-angles.
Thus each side is the hypotenuse of a right triangle with legs equal to the half-diagonals.
Thus if s=length of <i>each</i> side of the rhombus, then
s²=(AC/2)²+(BD/2)²
using Pythagoras theorem.
Calculate
s=√(7²+24²)
=25
The perimeter is 4s.
In that case, we make use of the fact that rhombuses have diagonals which intersect at right-angles.
Thus each side is the hypotenuse of a right triangle with legs equal to the half-diagonals.
Thus if s=length of <i>each</i> side of the rhombus, then
s²=(AC/2)²+(BD/2)²
using Pythagoras theorem.
Calculate
s=√(7²+24²)
=25
The perimeter is 4s.
Answered by
AoPS
Stop cheating Sheenybeany!
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