+0

help

0
52
1

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

#1
+8406
+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
+8406
+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

hectictar Jun 18, 2019