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The square with vertices (-1, -1), (1, -1), (-1, 1) and (1, 1) is cut by the line \(y=\frac{x}{2}+ 1\) into a triangle and a pentagon. What is the number of square units in the area of the pentagon? Express your answer as a decimal to the nearest hundredth.

 Jun 18, 2019

Best Answer 

 #1
avatar+8842 
+2

 

area of pentagon   =   area of square - area of triangle

 

area of square   =   (side length)2   =   22   =   4

 

To find the area of the triangle, we need to find the coordinates of  A  and  B .

 

To find point  A , we need to find the y-coordinate of the line when  x = -1 .

 

\(y\ =\ \frac{x}{2}+1\\~\\y\ =\ \frac{-1}{2}+1\\~\\y\ =\ \frac12\)

 

So   A  =  (-1, \(\frac12\))

 

To find point  B , we need to find the x-coordinate of the line when  y = 1 .

 

\(y\ =\ \frac{x}{2}+1\\~\\ 1\ =\ \frac{x}{2}+1\\~\\ 0\ =\ \frac{x}{2}\\~\\ 0\ =\ x\)

 

So   B  =  (0, 1)

 

base of triangle   =   0 - -1   =   1

 

height of triangle  =  1 - \(\frac12\)   =   \(\frac12\)

 

area of triangle   =   \(\frac12\)( base )( height )   =   \(\frac12\)( 1 )( \(\frac12\) )   =   \(\frac14\)

 

Now we can find the area of the pentagon.

 

area of pentagon   =   area of square - area of triangle   =   4 - \(\frac14\)   =   3.75

 

Here's the graph:  https://www.desmos.com/calculator/prypeui4fb

 Jun 18, 2019
 #1
avatar+8842 
+2
Best Answer

 

area of pentagon   =   area of square - area of triangle

 

area of square   =   (side length)2   =   22   =   4

 

To find the area of the triangle, we need to find the coordinates of  A  and  B .

 

To find point  A , we need to find the y-coordinate of the line when  x = -1 .

 

\(y\ =\ \frac{x}{2}+1\\~\\y\ =\ \frac{-1}{2}+1\\~\\y\ =\ \frac12\)

 

So   A  =  (-1, \(\frac12\))

 

To find point  B , we need to find the x-coordinate of the line when  y = 1 .

 

\(y\ =\ \frac{x}{2}+1\\~\\ 1\ =\ \frac{x}{2}+1\\~\\ 0\ =\ \frac{x}{2}\\~\\ 0\ =\ x\)

 

So   B  =  (0, 1)

 

base of triangle   =   0 - -1   =   1

 

height of triangle  =  1 - \(\frac12\)   =   \(\frac12\)

 

area of triangle   =   \(\frac12\)( base )( height )   =   \(\frac12\)( 1 )( \(\frac12\) )   =   \(\frac14\)

 

Now we can find the area of the pentagon.

 

area of pentagon   =   area of square - area of triangle   =   4 - \(\frac14\)   =   3.75

 

Here's the graph:  https://www.desmos.com/calculator/prypeui4fb

hectictar Jun 18, 2019

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