PLEASE ANSWER THIS

Question

ODEF is a square. In terms of a, find -point e is (2a, 2a). point f is (0, 2a). point o is (0,0) and point d is (2a, 0) 2. The area of the square 3. The midpoint of FD 4. The midpoint of OE Point O= (0,0) Point D= (2a,0)

Answer 1

Since they gave you all four corners as coordinates, we can do this whole problem with the coordinate formulas instead of trying to picture it, and that is usually the easier way to go on these. Our four points are O(0, 0), D(2a, 0), E(2a, 2a) and F(0, 2a).

For the area, we first need one side length. Looking at O(0, 0) and D(2a, 0), both points have the same y value, so that side is horizontal and we can just subtract the x values to get 2a - 0 = 2a. So the side of the square is 2a, and since area of a square is side times side:

Area = (2a)² = 4a²

If you notice, the 2 gets squared too, not just the a, which is the most common mistake on this one. Students very often write 2a², but (2a)² means 2a times 2a, and that is 4a².

For the midpoints we use the midpoint formula, where you just average the x values and average the y values:

midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2)

So for FD, with F(0, 2a) and D(2a, 0):

((0 + 2a)/2, (2a + 0)/2) = (a, a)

And for OE, with O(0, 0) and E(2a, 2a):

((0 + 2a)/2, (0 + 2a)/2) = (a, a)

Now here is the part I want you to catch, because it is the whole point of the problem. Both midpoints came out to the exact same point, (a, a). That is not a coincidence and it is not a mistake in your work. FD and OE are the two diagonals of the square, and one of the properties of a square is that its diagonals bisect each other, which means they cross at their shared middle. So both of them have to have the same midpoint, and that point is the center of the square.

A nice way to check this is to notice that (a, a) is exactly halfway across the square both left to right and bottom to top, since the square runs from 0 to 2a in both directions. So the center landing at (a, a) is exactly what we should expect.

One more thing that falls right out of this if your problem asks for it later. The diagonals both have length:

√((2a)² + (2a)²) = √(8a²) = 2a√2

and they are also perpendicular to each other, which is another property of a square specifically. A rectangle has diagonals that bisect each other but are not perpendicular, so that is the detail that separates the two.

Also, your list starts at number 2, so it looks like part 1 got cut off when you posted. If part 1 was asking for the side length or the perimeter, the side is 2a and the perimeter is 8a. Just let me know if it was something else and I can go through that part too.

So, our final answers are 4a² for the area, (a, a) for the midpoint of FD, and (a, a) again for the midpoint of OE.