Parallelogram 65334

In a parallelogram, the sum of the lengths of the sides a+b = 234. The angle subtended by the sides a and b is 60°. The diagonal size against the given angle of 60° is u=162. Calculate the sides of the parallelogram, its perimeter, and its area.

Correct answer:

a =  181.6916
b =  52.3084
o =  468
S =  8230.7054

Step-by-step explanation:

$$\begin{gathered} a+b=234 \\ \beta =60°=60 \\ u=162 \\ {u}^2=a^2+b^2-2\cdot a\cdot b\cdot \cos \beta \\ {u}^2=a^2+(234-a{)}^2-2\cdot a\cdot (234-a)\cdot 0.5 \\ 162^2=a^2+(234-a{)}^2-2\cdot a\cdot (234-a)\cdot 0.5 \\ -3a^2+702a-28512=0 \\ 3a^2-702a+28512=0 \\ 3...\text{ prime number} \\ 702=2\cdot 3^3\cdot 13 \\ 28512=2^5\cdot 3^4\cdot 11 \\ \text{GCD}(3,702,28512)=3 \\ {a}^2-234a+9504=0 \\ p=1;q=-234;r=9504 \\ D=q^2-4pr=234^2-4\cdot 1\cdot 9504=16740 \\ D>0 \\ {a}_{1,2}=\frac{-q\pm \sqrt{D}}{2p}=\frac{234\pm \sqrt{16740}}{2}=\frac{234\pm 6\sqrt{465}}{2} \\ {a}_{1,2}=117\pm 64.691576 \\ {a}_1=181.691575959 \\ {a}_2=52.308424041 \\ a=a_1=181.6916 \end{gathered}$$

Our quadratic equation calculator calculates it.

$b=234-a=234-181.6916=52.3084$

Try calculation via our triangle calculator.

$o=2\cdot (a+b)=2\cdot (181.6916+52.3084)=468$

$$\begin{gathered} \sin \beta =v:b \\ v=b\cdot \sin \beta =b\cdot \sin 60°=52.3084\cdot \sin 60°=52.3084\cdot 0.866025=45.30042 \\ S=a\cdot v=181.6916\cdot 45.3004=8230.7054 \end{gathered}$$

Try another example