Question

Find the area of rectangle PLUM

If entering your answer as a decimal, round your final answer to the nearest hundredth.

Answer 1

A = 53.34
`units ^ 2`

Explanation:

To find the length of the rectangle we solve the PLA triangle

Using the Pythagorean theorem we find the PL side.

`PL = √(4^2+12^2)\\PL = √(16+144)\\PL = 12.65`

Then the length of the rectangle is 12.65

Now we solve the MALU triangle. Where MU = PL

We use the Pythagorean theorem:

`MU^2+LU^2 = (MA + AL)^2\\12.65^2+LU^2 = (MA+12)^2`

Where LU and MA are unknown.

Finally we solve the triangle MPA

`MA^2 + PA^2 = PM ^2`

Where PM = LU

So:

`MA^2 + 4^2 = LU^2`

Now we have two equations and two unknowns. So we solve the system.
`12.65^2 + LU^2 = (MA + 12) ^2` (i)

`MA^2+4^2 = LU^2` (ii)

We introduce (ii) in (i)

`12.65^2 + MA^2+16=MA^2+24MA+144`

Now we clear MA

`160.02+16-1 44 = 24MA`

`32.02 = 24MA`

`MA = (32.02)/(24)`

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
`units^2`