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Hi All,
I would grateful if someone provided an explanation to the problem below. I provided the OG explanation and it just doesn't click.
The diagram shows a parallelogram which has four sides of equal length and an angle equal to 60°
The parallelogram shown has four sides of equal length. What is the ratio of the length of the shorter diagonal to the length of the longer diagonal?
Solution
First, the opposite angles of a parallelogram have equal measure, and the sum of the measures of adjacent angles is 180°. This means that each of the angles adjacent to the angle labeled 60° has measure 120°.
Since all four sides of the parallelogram have equal length, say x units, the shorter diagonal divides the parallelogram into two isosceles triangles. An isosceles triangle with one angle measuring 60° is equilateral, and so the shorter diagonal has length x units.
The longer diagonal divides the parallelogram into two isosceles triangles with one angle measuring 120° and each of the other angles measuring
(180-120)/2 = 30°.
Then, because the diagonals of a parallelogram are perpendicular and bisect each other, the two diagonals divide the parallelogram into four 30°-60°-90° triangles, each with hypotenuse x units long. The sides of a 30°-60°-90° triangle are in the ratio of 1:√3:2 and so, if y represents the length of the side opposite the 60° angle, then x/2 = y/√3 and y = (x√3)/2. But, y is half the length of the longer diagonal, so the longer diagonal has length x√3 units. Therefore, the ratio of the length of the shorter diagonal to the length of the longer diagonal is x/(x√3)=1/√3.
I would grateful if someone provided an explanation to the problem below. I provided the OG explanation and it just doesn't click.
The diagram shows a parallelogram which has four sides of equal length and an angle equal to 60°
The parallelogram shown has four sides of equal length. What is the ratio of the length of the shorter diagonal to the length of the longer diagonal?
Solution
First, the opposite angles of a parallelogram have equal measure, and the sum of the measures of adjacent angles is 180°. This means that each of the angles adjacent to the angle labeled 60° has measure 120°.
Since all four sides of the parallelogram have equal length, say x units, the shorter diagonal divides the parallelogram into two isosceles triangles. An isosceles triangle with one angle measuring 60° is equilateral, and so the shorter diagonal has length x units.
The longer diagonal divides the parallelogram into two isosceles triangles with one angle measuring 120° and each of the other angles measuring
(180-120)/2 = 30°.
Then, because the diagonals of a parallelogram are perpendicular and bisect each other, the two diagonals divide the parallelogram into four 30°-60°-90° triangles, each with hypotenuse x units long. The sides of a 30°-60°-90° triangle are in the ratio of 1:√3:2 and so, if y represents the length of the side opposite the 60° angle, then x/2 = y/√3 and y = (x√3)/2. But, y is half the length of the longer diagonal, so the longer diagonal has length x√3 units. Therefore, the ratio of the length of the shorter diagonal to the length of the longer diagonal is x/(x√3)=1/√3.
