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A rectangle in the coordinate plane is shown below. A line with slope  1/3 splits this rectangle into two regions with equal area. What is the y-intercept of this line?

(-2,3)  ____________    (4,3)
           |                       |
           |                       |
           |___________ |

(-2, 5)                                    (4, -5)

 

Note: The graph is not to scale

 Apr 7, 2024
 #1
avatar+9664 
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I think you made a typo, the bottom-left corner is likely to be (-2, -5) instead of (-2, 5).

 

Key observation: If a line bisects the area of a rectangle, it must pass through the intersection of diagonals, i.e., the centre of the rectangle.

(It will divide the rectangle into two identical trapezoids that way.)

 

We first find the intersection of diagonals, i.e., the mid-point of a diagonal. That point is \(\left(\dfrac{-2 + 4}2, \dfrac{5 + 3}2\right) = (1, 4)\).

The line has slope 1/3 and passes through (1, 4). We can find the equation of the line through "point-slope form" of straight line equations.

The equation is 

\(\dfrac{y - 4}{x - 1} = \dfrac13\\ y - 4 = \dfrac13 x - \dfrac13\\ y = \dfrac13 x + \dfrac{11}3\)

 

Now that the equation is in slope-intercept form, we can directly read off the y-intercept from the equation. 

The y-intercept of the line is \(\dfrac{11}3\).

 Apr 8, 2024
 #2
avatar+39 
0

The bottom left corner is (-2,-5)! I just checked

rerebas  Apr 10, 2024

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