A triangle has vertices a(0,2),b(3,7)and c(0,6)give that abcd is parallelogram find the area of parallelogram abcd
we could use CA as the base of the triangle ABC
then CA = 4
we could use the distance of B(3,7) from the y-axis as the height
height = 3
area of triangle = (1/2)(4)(3) = 6 units^2
so the area of the parallelogram is 12 units^2
To find the area of the parallelogram ABCD, we can use the formula:
Area = base x height
In this case, the base of the parallelogram would be the length of side AB, and the height would be the distance from side AB to the opposite side. Since ABCD is a parallelogram, the opposite side to AB would be the side that is parallel to it, which is side CD.
To find the length of side AB, we can use the distance formula:
Distance = √((x2 - x1)^2 + (y2 - y1)^2)
In this case, the coordinates of point A are (0,2) and the coordinates of point B are (3,7). Plugging these values into the distance formula, we can find the length of side AB.
Distance_AB = √((3 - 0)^2 + (7 - 2)^2)
Distance_AB = √(3^2 + 5^2)
Distance_AB = √(9 + 25)
Distance_AB = √34
Now, we need to find the distance from side AB to side CD. Since side AB is parallel to side CD, the height of the parallelogram would be the perpendicular distance between side AB and side CD. We can use the formula for the distance between a point and a line to find this distance.
The equation of the line passing through points C(0,6) and D(x,y) can be found using the slope-intercept form (y = mx + b) where m is the slope and b is the y-intercept.
The slope m can be found using the formula:
m = (y2 - y1) / (x2 - x1)
In this case, the coordinates of point C are (0,6) and the coordinates of point D are (x,y). Plugging these values into the slope formula, we get:
m = (y - 6) / (x - 0)
m = (y - 6) / x
Since ABCD is a parallelogram, the slope of side CD would be equal to the slope of side AB. Therefore, we can set the two slope equations equal to each other:
(y - 6) / x = (7 - 2) / (3 - 0)
(y - 6) / x = 5 / 3
Cross-multiplying, we get:
3(y - 6) = 5x
3y - 18 = 5x
3y = 5x + 18
y = (5/3)x + 6
Now we have the equation of line CD. We can find the perpendicular distance between AB and CD by finding the distance between point C and line CD using the formula:
Distance_CD = |Ax + By + C| / √(A^2 + B^2)
Where A, B, and C are coefficients of the equation of line CD (y = (5/3)x + 6). In this case, A = 5/3, B = -1, and C = -18. Plugging these values into the formula:
Distance_CD = |(5/3)(0) + (-1)(6) + (-18)| / √((5/3)^2 + (-1)^2)
Distance_CD = 24 / √(25/9 + 1)
Distance_CD = 24 / √(34/9 + 9/9)
Distance_CD = 24 / √(43/9)
Distance_CD = (24 * √9) / √43
Distance_CD = 8√43 / 3
Now, we have calculated the length of side AB (Distance_AB = √34) and the height of the parallelogram (Distance_CD = 8√43 / 3).
Finally, we can calculate the area of the parallelogram using the formula:
Area = base x height
Area = Distance_AB x Distance_CD
Area = √34 x (8√43 / 3)
Area = (8/3)√(34 * 43)
Area = (8/3)√1462
Therefore, the area of parallelogram ABCD is (8/3)√1462.