Answer:
A = 53.34 [tex]units ^ 2[/tex]
Step-by-step explanation:
To find the length of the rectangle we solve the PLA triangle
Using the Pythagorean theorem we find the PL side.
[tex]PL = \sqrt{4^2+12^2}\\PL = \sqrt{16+144}\\PL = 12.65[/tex]
Then the length of the rectangle is 12.65
Now we solve the MALU triangle. Where MU = PL
We use the Pythagorean theorem:
[tex]MU^2+LU^2 = (MA + AL)^2\\12.65^2+LU^2 = (MA+12)^2[/tex]
Where LU and MA are unknown.
Finally we solve the triangle MPA
[tex]MA^2 + PA^2 = PM ^2[/tex]
Where PM = LU
So:
[tex]MA^2 + 4^2 = LU^2[/tex]
Now we have two equations and two unknowns. So we solve the system.[tex]12.65^2 + LU^2 = (MA + 12) ^2[/tex] (i)
[tex]MA^2+4^2 = LU^2[/tex] (ii)
We introduce (ii) in (i)
[tex]12.65^2 + MA^2+16=MA^2+24MA+144[/tex]
Now we clear MA
[tex]160.02+16-1 44 = 24MA[/tex]
[tex]32.02 = 24MA[/tex]
[tex]MA = \frac{32.02}{24}[/tex]
MA = 1.3342 (iii)
Now we introduce (iii) in (ii)
1.3342^2 + 4^2 = LU^2
LU = 4.2166
Now we have the length of the rectangle and also its width.
The area A of a rectangle is:
A = l*w
Where
l = length
w = width
A = 4.2166*12.65
A = 53.34 [tex]units^2[/tex]
