+0  
 
0
185
2
avatar

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

Best Answer 

 #2
avatar+81057 
+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
Sort: 

2+0 Answers

 #1
avatar
0

Correction : 37°25' sorry :D

Guest Jan 4, 2015
 #2
avatar+81057 
+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

18 Online Users

avatar
We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners.  See details