Question

Three coordinate points of a parallelogram are (2, 1), (4, 4), and (7, 4). Find the fourth vertex

Answer 1

The fourth vertex of the parallelogram with vertices at (2, 1), (4, 4), and (7, 4) is found to be (5, 4) using vector or midpoint techniques. The calculation involves ensuring the opposite sides of the parallelogram are parallel and of equal length.

Step-by-step explanation:

To find the fourth vertex of a parallelogram given three coordinate points, we use the concept that opposite sides of a parallelogram are equal in length and parallel. Let's name the given points as A(2, 1), B(4, 4), and C(7, 4). To find the fourth vertex, which we'll call point D, we can use the vector approach or the midpoint formula.

Let us assume that A and C are opposite vertices, and B is one of the adjacent vertices to A. The vector from A to B is (B - A), which is (4 - 2, 4 - 1) = (2, 3). Since AC is a diagonal, the fourth point D should be such that vector AD equals vector BC. If D has coordinates (x, y), then vector AD = (x - 2, y - 1). Equating this to vector BC (which is 0 since B and C have the same y-coordinate), we get x - 2 = 0 and y - 1 = 3. Therefore, x = 2 and y = 4. Point D would then be (2 + 3, 1 + 3) = (5, 4).

Thus, the coordinates of the fourth vertex of the parallelogram are (5, 4).

Answer 2

To find the fourth vertex of the parallelogram, use the fact that opposite sides of a parallelogram are equal in length and parallel to each other. Find the distance between two given points to determine the length of one side, and then use this length and the coordinates of one of the given points to find the coordinates of the fourth vertex.

Step-by-step explanation:

To find the fourth vertex of the parallelogram, we can use the fact that opposite sides of a parallelogram are equal in length and parallel to each other. First, find the distance between the points (2, 1) and (4, 4) to determine the length of one side of the parallelogram. Then, use this length and the coordinates of one of the given points to find the coordinates of the fourth vertex.

Distance between (2, 1) and (4, 4) = sqrt((4 - 2)^2 + (4 - 1)^2) = sqrt(2^2 + 3^2) = sqrt(13)

Let's say the fourth vertex has coordinates (x, y). Since opposite sides of a parallelogram are parallel, the distance between (x, y) and (2, 1) should also be sqrt(13). So, we have sqrt((x - 2)^2 + (y - 1)^2) = sqrt(13).

Solving this equation, we get (x - 2)^2 + (y - 1)^2 = 13. We can choose any value for x or y and find the corresponding value for the other variable. For example, let's set x = 2. Substituting this into the equation, we get (2 - 2)^2 + (y - 1)^2 = 13, which simplifies to (y - 1)^2 = 13. Taking the square root of both sides gives us y - 1 = sqrt(13) or y - 1 = -sqrt(13). Solving for y, we get y = 1 + sqrt(13) or y = 1 - sqrt(13). Therefore, the fourth vertex of the parallelogram can either be (2, 1 + sqrt(13)) or (2, 1 - sqrt(13)).