The area of a rhombus is 250 and one of the angles is 37°27'. What is the length of each side?

Guest Jan 4, 2015

#2**+5 **

This one is a little trcky......

Call the longer diagonal D1 and the shorter one D2

And the area of the rhombus = (1/2) the product of the diagonals...so we have

250 = (1/2) (D1)(D2) → (D1)(D2) = 500 → D2 = 500/D1

And adjacent angles in a rhombus are supplemental....so, the other angle is = (108- 37.25) = 142.75°

And the greater angle lies opposite its respective diagonal, D1...and the lesser angle lies opposite the shorter diagonal, D2 ...and the diagonals bisect both of these angles

So...using the Law of Sines, we have

sin(37.25/2) / (1/2)(D2) = sin(142.75/2)/(1/2)D1 and substituting for D2, we have....

sin(37.25/2) / (1/2)(500/D1) = sin(142.75/2)/(1/2)D1 simplify...

(D1)sin(18.625) / 500 = sin(71.375) / D1

(D1)^2 = 500sin(71.375)/sin(18.625) = 1483.5782595004849666 take the square root of both sides

D1 = about 38.5 and D2 = 500/D1 = 500/38.5 = about 12.99

And we can find the side length - S - using the Pythagorean Theorem....(1/2) of the length of each diagonal will be the "leg" lengths, and the hypoteneuse will be the side length

So we have

S = √(19.25^2 + 6.495^2) = S = about 20.3...and that's the side length...!!!

CPhill
Jan 4, 2015

#2**+5 **

Best Answer

This one is a little trcky......

Call the longer diagonal D1 and the shorter one D2

And the area of the rhombus = (1/2) the product of the diagonals...so we have

250 = (1/2) (D1)(D2) → (D1)(D2) = 500 → D2 = 500/D1

And adjacent angles in a rhombus are supplemental....so, the other angle is = (108- 37.25) = 142.75°

And the greater angle lies opposite its respective diagonal, D1...and the lesser angle lies opposite the shorter diagonal, D2 ...and the diagonals bisect both of these angles

So...using the Law of Sines, we have

sin(37.25/2) / (1/2)(D2) = sin(142.75/2)/(1/2)D1 and substituting for D2, we have....

sin(37.25/2) / (1/2)(500/D1) = sin(142.75/2)/(1/2)D1 simplify...

(D1)sin(18.625) / 500 = sin(71.375) / D1

(D1)^2 = 500sin(71.375)/sin(18.625) = 1483.5782595004849666 take the square root of both sides

D1 = about 38.5 and D2 = 500/D1 = 500/38.5 = about 12.99

And we can find the side length - S - using the Pythagorean Theorem....(1/2) of the length of each diagonal will be the "leg" lengths, and the hypoteneuse will be the side length

So we have

S = √(19.25^2 + 6.495^2) = S = about 20.3...and that's the side length...!!!

CPhill
Jan 4, 2015