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Twelve 1 by 1 squares form a rectangle, as shown. What is the total area of the shaded region?

 Jul 17, 2019
 #1
avatar+23510 
+4

Twelve 1 by 1 squares form a rectangle, as shown. What is the total area of the shaded region?

 

\(\text{Let the area of the lower triangle is $\dfrac{2\cdot 4}{2} = 4 $ }\\ \text{Let the area of the upper triangle is $\dfrac{3\cdot 4}{2} = 6 $ }\\ \text{The area of the shaded region is $4+6=\mathbf{10} $ }\)

 

laugh

 Jul 17, 2019
edited by heureka  Jul 17, 2019
edited by heureka  Jul 17, 2019
 #2
avatar+105195 
+3

Note that the shaded area is composed of two triangles...

 

The one on the left has a height of 4 and a base of 2.....so its area  = (1/2)(2)(4) =  4 units^2

 

The one on the right has a base of 3 and a height of 4....so its area  = (1/2)(3)(4)  = 6 units ^2

 

So.....the total shaded area  is  [ 4 + 6 ] iunits^2  = 10 units^2

 

 

cool cool cool

 Jul 17, 2019
 #3
avatar
+1

Slightly different approach - let's subtract off the non-shaded region.

 

By Pick's Theorem, we have that the area is $$\frac{4}{2}+1 - 1 = 2.$$ Note that this only works if the vertices are lattice points and there are no crosses in the figure. Then, the area of the rectangle is \(4\times3\), so our final answer is $$4\times3 - 2 = \boxed{10}.$$

.
 Jul 18, 2019

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