What is the area of the quadrilateral bounded by #y=5#; #x=1#; #y=1#; and #y=-2x+9#?

Question

Aaliyah Anderson · Accepted Answer

#8# square units

Note that #y=-2x+9# can be written as #x=(9-y)/2#
making it easier to evaluate the intersection coordinates for
#y=-2x+9# and #y=5#
#color(white)(y=-2x+9)# and #y=1#

From the above diagram it is clear that the area is

#color(white)("XXX")color(blue)4 xx (color(blue)1 + color(blue)3)/2#

#color(white)("XXX")=8#

Aaliyah Anderson · Answer

undefined To find the area of the quadrilateral bounded by the given equations, we need to first identify the vertices of the quadrilateral by finding their intersection points. Then, we can use the formula for finding the area of a quadrilateral formed by four given points.

1. Find the intersection points of the given equations:
   - For y = 5 and x = 1: The intersection point is (1, 5).
   - For y = 1 and x = 1: The intersection point is (1, 1).
   - For y = -2x + 9 and y = 1:
     Substitute y = 1 into y = -2x + 9:
     1 = -2x + 9
     Solving for x: x = 4.
     Substitute x = 4 into y = -2x + 9: y = 1.
     The intersection point is (4, 1).
   - For y = -2x + 9 and y = 5:
     Substitute y = 5 into y = -2x + 9:
     5 = -2x + 9
     Solving for x: x = 2.
     Substitute x = 2 into y = -2x + 9: y = 5.
     The intersection point is (2, 5).

2. Now, we have the vertices of the quadrilateral: (1, 5), (1, 1), (4, 1), and (2, 5).
   
3. Calculate the area of the quadrilateral using these vertices. We can divide the quadrilateral into two triangles and find the area of each triangle using the formula for the area of a triangle (0.5 * base * height), then sum the areas of the triangles.

Triangle 1: Vertices (1, 5), (1, 1), and (4, 1).
   - Base = Distance between (1, 5) and (4, 1) = √((4-1)^2 + (1-5)^2) = √(3^2 + (-4)^2) = √(9 + 16) = √25 = 5 units
   - Height = Distance between (1, 1) and (1, 5) = 5 - 1 = 4 units
   - Area = 0.5 * base * height = 0.5 * 5 * 4 = 10 square units

Triangle 2: Vertices (1, 1), (4, 1), and (2, 5).
   - Base = Distance between (1, 1) and (2, 5) = √((2-1)^2 + (5-1)^2) = √(1^2 + 4^2) = √(1 + 16) = √17 units
   - Height = Distance between (1, 1) and (4, 1) = 4 - 1 = 3 units
   - Area = 0.5 * base * height = 0.5 * √17 * 3 = 1.5√17 square units

4. Total area of the quadrilateral = Area of Triangle 1 + Area of Triangle 2 = 10 + 1.5√17 square units.