Find the area of a trapezoid ABCD with verticles A(2,2) B(4,6) C(4,-3)
and D(2,-1).
Should I use a graph and go from there? I'm just not sure of how to set this up. The book answer is 12units. I am not coming up with the answer
A Trapezoid has 2 parallel sides which
have equal slopes, and 2 non-parallel
sides with unequal slopes. So we calculate the slope of all 4 sides and
make comparisons:
AB. m = (6-2) / (4-2) = 4/2 = 2.
BC. m=(-3-6) / (4-4)=-9/0 = undefined.
CD. m = (-1-(-3)) / (2-4) = 2/-2 = -1.
AD. m=(-1-2) / (2-2)=-3/0 = undefined.
The 2 lines with the undefined slopes
are parallel. The other 2 are non-para-
llel. The parallel lines are normally
horizontal with a slope of zero. The
trapezoid in this prob. has been rotated 90 degrees which accounts for
the undefined slopes. The slopes are
equal to tangent of the angle:
tanA = 2. A = 63.4 deg.
(AB)^2 = (4-2)^2 + (6-2)^2 = 4+16 = 20,
AB = 4.47.
h = 4.47sin63.4 = 4.
tanE = -1. D = 180-135 = 45 deg.
E = exterior angle. D = interior angle.
CD = h / sinD = 4 / sin45 = 5.66.
(BC)^2 = (4-4)^2 + (-3-6)^2 = 81,
BC = 9.
(AD)^2 = (2-2)^2 + (-1-2)^2 = 9,
AD = 3.
Area = (BC + AD)h/2 = (9+3)4 / 2 = 24.
My answer is twice your book's answer.
Please make sure your book is correct.
To find the area of a trapezoid, you can use the formula A = ((b1 + b2) * h) / 2, where b1 and b2 are the lengths of the parallel sides (bases) and h is the height of the trapezoid.
First, let's identify the bases and the height of the trapezoid using the given coordinates.
The bases b1 and b2 are the lengths of the parallel sides AB and CD, respectively. You can calculate the lengths of these sides using the distance formula:
Distance formula: d = √((x2 - x1)^2 + (y2 - y1)^2)
The height h is the perpendicular distance between the bases AB and CD. To find the height, you need to find the equation of the line passing through points B(4,6) and C(4,-3). Since this line is vertical, its equation will be x = 4.
Let's calculate the lengths of the bases AB and CD:
AB = √((4 - 2)^2 + (6 - 2)^2)
= √(2^2 + 4^2)
= √(4 + 16)
= √20
= 2√5 (approximately 4.472 units)
CD = √((4 - 2)^2 + (-3 - (-1))^2)
= √(2^2 + (-3 + 1)^2)
= √(4 + 4)
= √8
= 2√2 (approximately 2.828 units)
Now, let's calculate the height h:
The equation of the vertical line passing through point B(4,6) is x = 4.
Since vertices A(2,2) and D(2,-1) lie on this line, the height is the difference between their y-coordinates:
h = 2 - (-1)
= 3 units
Now, substitute the values of the bases and the height into the formula for the area of a trapezoid:
A = ((b1 + b2) * h) / 2
= ((2√5 + 2√2) * 3) / 2
= (6√5 + 6√2) / 2
= 3√5 + 3√2
The calculated area of the trapezoid is 3√5 + 3√2, which is approximately 10.04 units.
However, the book answer is given as 12 units. Therefore, it appears that there might be an error either in the given coordinates or the book answer.