Question

Two smiddle" class="latex-formula"> rectangles are drawn, as shown. Find the area of their overlap, which is shaded.

Answer 1

Answer:

$\huge\boxed{A=3\dfrac{1}{3}}$

Step-by-step explanation:

Look at the picture.

Shadded region is a parallelogram.

The formula of an area of a prarallelogram:

$A=bh\\\\b-base\\h-height$

We have the height h = 2.

Calculate the base.

Green triangles are congruent.

Use the Pytgagorean theorem:

$leg^2+leg^2=hypotenuse^2$

We have:

$leg=x\\leg=2\\hypotenuse=3-x$

Substitute:

$x^2+2^2=(3-x)^2\qquad\text{use}\ (a-b)^2=a^2-2ab+b^2\\\\x^2+4=3^2-2(3)(x)+x^2\qquad\text{subtract}\ x^2\ \text{from both sides}\\\\4=9-6x\qquad\text{subtract 9 from both sides}\\\\-5=-6x\qquad\text{divide both sides by (-6)}\\\\\dfrac{-5}{-6}=x\to x=\dfrac{5}{6}$

Therefore the base is:

$b=3-x\to b=3-\dfrac{5}{6}=\dfrac{18}{6}-\dfrac{5}{6}=\dfrac{13}{6}$

Calculate the area:

$A=\left(\dfrac{13}{6}\right)(2)=\dfrac{26}{6}=\dfrac{13}{3}=4\dfrac{1}{3}$