A quadrilateral is formed by the four lines where y=5, x=8, y=2x-5, and y= -3. Find the area of the quadrilateral
Since the area is bounded by two parallel lines y=5 and y=-3, the area is a trapezoid, with a height of (5-(-3))=8.
We need to find the intersection of the line y=2x-5 with each of the parallel lines.
Setting 5=2x-5 => x=5
Setting -3=2x-5 => x=1
So the upper width is (8-5)=3
and the lower width is (8-1)=7
The area of the trapezoid is therefore
Area = (1/2)(3+7)*8
=40 units.
To find the area of the quadrilateral formed by these four lines, we can first identify the coordinates of the four points where these lines intersect.
1. Line y=5 intersects with x=8 at the point (8, 5).
2. Line x=8 intersects with y=2x-5 by substituting x=8 into y=2x-5. Therefore, y=2(8)-5=11. So, (8, 11) is the intersection point of these two lines.
3. Line y=2x-5 intersects with y=-3 by substituting y=-3 into y=2x-5. Therefore, -3=2x-5. Solving for x gives x=1. So, (1, -3) is the intersection point of these two lines.
4. Line y=-3 intersects with y=5 by substituting y=5 into y=-3. Therefore, 5=-3, which is not possible.
Now we have three points — (8, 5), (8, 11), and (1, -3) — to form a triangle. The base of this triangle will be the difference in x-coordinates between the points (8, 5) and (1, -3), which is 8 - 1 = 7.
Next, we need to find the height of the triangle. The height is the perpendicular distance from the point (8, 11) to the line connecting the other two points.
Using the formula for the equation of a line in slope-intercept form (y = mx + b), the slope of the line connecting the points (8, 5) and (1, -3) is (5 - (-3))/(8 - 1) = 8/7.
The perpendicular slope to this line will be the negative reciprocal of 8/7, which is -7/8.
Using the point-slope form (y - y1) = m(x - x1), where (x1, y1) = (8, 11) and m = -7/8, we substitute these values into the equation. We get y - 11 = (-7/8)(x - 8), which simplifies to y = (-7/8)x + 91/8.
Now, we need to find the y-coordinate of the point (8, 11) on this line. Substituting x = 8 into the equation, we have y = (-7/8)(8) + 91/8 = -7 + 91/8 = 35/8.
Finally, the height of our triangle is the difference between the y-coordinate of the point (8, 11) and the y-coordinate of the point (8, 5), which is (35/8) - 5 = (35-40)/8 = -5/8.
Now we have the base (7) and the height (-5/8) of the triangle. To calculate the area, we use the formula for the area of a triangle: Area = (1/2) * base * height.
Plugging in the values, we get Area = (1/2) * 7 * (-5/8). Simplifying, we have Area = -35/16.
Therefore, the area of the quadrilateral formed by the lines y=5, x=8, y=2x-5, and y=-3 is -35/16 square units.