In a rectangle the angle between diagonals is 60°. The sum of the lengths of both diagonals and both shorter sides of the rectangle is 36 in. What are the lengths of the diagonals?

so the diagonal and the sides of the rectangle will form a right triangle. where the the angle are 30 60 90 degrees, this is a special triangle because the shortest leg and the hypotenuse has a ratio of 1 : 2, meaning the diagonal and the shorter leg is in the same ratio. so let x be the length of the diagonal and y be the length of the shorter side.

x + y = 36

since x = 2y
2y + y = 36
3y = 36
y = 12

x = 24 in

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vintechnology vintechnology · Answer 2 · 2022-03-15T10:02:11-04:00

The length of the diagonals are 12 inches

Properties of a rectangle:

The length of the two diagonal are equal
Opposite sides are equal
Diagonals are of equal length
Diagonal are bisector of each other

Therefore, the angle where the diagonal intersect are 60 degrees each. Therefore,

AOB = 60 degrees

DOC = 60 degrees

where

O is the centre where they intersect

The triangles are equilateral triangles. Therefore, the sides length are the same.

shorter sides = 2x

Length of the two diagonal = 2x + 2x = 4x

Therefore,

2x + 4x = 36

6x = 36

x = 36 / 6

x = 6

The length of the diagonals are 2(x) = 2(6) = 12 inches

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