Question
In quadrilateral ABCD, we have AB = BC = CD = DA, AC = 14, and BD = 48. Find the perimeter of ABCD.
Answers
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.
Sheenybeany
Yeah, but we're not trying to find the area?
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.
AoPS
Stop cheating Sheenybeany!