(i) AB and DC are parallel, so the lines containing these sides have the same slope.
The line containing AB has slope (2 - (-1))/(10 - 4) = 3/6 = 1/2.
Use the point-slope formula to find the equation of the line containing CD:
AP is perpendicular to both AB and CD, and perpendicular lines have slopes that are negative reciprocals of one another. This means the line containing AP has slope -1/(1/2) = -2. Using the point-slope formula again, the equation of the line containing AP is
(ii) D lies on the x axis, so its y-coordinate is 0. Find the x-coordinate by plugging y = 0 into the equation for the line containing CD:
So D is the point (-4, 0).
P is the intersection of the lines containing CD and AP, so set them equal and solve for x and y :
So P is the point (2, 3).
(iii) For ASBC to be a parallelogram, we need to find the coordinates of S such that the line containing AS is parallel to the line containing BC, and the line containing SC is parallel to the line containing AB.
BC has slope (8 - 2)/(12 - 10) = 6/2 = 3. Then the line containing AS has equation
We already know AB has slope 1/2. Then the line containing SC has equation
S is the intersection of these two lines:
So S is the point (6, 5), and *not* (2, -7) as the answer key suggests. In fact, (2, -7) is located lower than the point A and slightly to the left; if you draw that point and connect it to the other three, there's no way to get a parallelogram.
(iv) Find the lengths of AB, CD, and AP. Then the area is (AB + CD)*AP/2.
so the area is
