my question in (distance between two point)
rectangle QRST has an area of 6 square units, and Q is at the origin. R lies on the x-axis, T lies on they-axis , and the coordinates of the vertices are whole numbers. what are the possible sets of coordinates of the rectangle ?
To find the possible sets of coordinates of the rectangle given its area and position of vertices, we need to use the formula for the area of a rectangle. The formula is:
Area = length × width
In this case, we know that the area is 6 square units. Let's assume the length of the rectangle is L and the width is W.
Since Q is at the origin (0,0), the coordinates of Q are (0,0). R lies on the x-axis, so the y-coordinate of R is 0. Let's say the x-coordinate of R is (x, 0).
Similarly, T lies on the y-axis, so the x-coordinate of T is 0. Let's say the y-coordinate of T is (0, y).
Now, we can determine the length and width of the rectangle. Since the opposite sides of a rectangle are equal, we have two scenarios:
Scenario 1: L = x and W = y
In this case, the area formula becomes:
6 = x × y
We need to find whole number values of x and y that satisfy the above equation. The possible combinations of x and y are:
1 × 6, 2 × 3, 3 × 2, and 6 × 1
So, the possible sets of coordinates for the rectangle are:
Q (0,0), R (1,0), S (1,6), T (0,6)
Q (0,0), R (2,0), S (2,3), T (0,3)
Q (0,0), R (3,0), S (3,2), T (0,2)
Q (0,0), R (6,0), S (6,1), T (0,1)
Scenario 2: L = y and W = x
In this case, the area formula becomes:
6 = y × x
Again, we need to find whole number values of x and y that satisfy the above equation. The possible combinations of x and y are the same as in scenario 1.
So, the possible sets of coordinates for the rectangle are the same as mentioned above.
To summarize, the possible sets of coordinates for the rectangle are:
Q (0,0), R (1,0), S (1,6), T (0,6)
Q (0,0), R (2,0), S (2,3), T (0,3)
Q (0,0), R (3,0), S (3,2), T (0,2)
Q (0,0), R (6,0), S (6,1), T (0,1)
area=R*T
6=RT
r,t = (1,6); (6,1); (3,2);(2,3)
check that.