Explanation



The two red ovals are the vertexes.

From this one side is 8 and the other as well

\(8 \times 8 = 64\)

C is the answer
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Hi,

How do you get point (10,10)?

Thanks

But the length of the square could be from (2,2) to (8,8) and the diagonal can still intersect with (6,6), wouldn't that be unable to determine the answer?

The given coordinates form a square as shown in the figure above.

Those must be the measures of the square.

Regards
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So how did you get coordinate (10,10)?

because one vertex point is (2,2).

If the diagonal intersection is at the midpoint (6,6), the other point is (10,10).

2 >> 4 >> 6 >> 4 >>10
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You can also do this...

Distance b/w point (2,2) and (6,6) using the formula.....root (X2 - X1)square + (Y2-Y1)square........it will come out to be 4root2.
therefore the length of the diagonal will be 8root2......

the triangle formed by the diagonal and 2 sides is a 45-45-90 right triangle therefore sides are in the ratio 1:1:root2..........therefore side comes out to be 8...

thus area of square is 8X8 = 64.

answer is C.

Cheers

We know that distance from (2,2) to (6,6) is 4 rise and 4 run. Using the pythagorean theorem we can say that the hypotenuse is √32=4√2.

The diagonal of the square therefore would be 8√2 (4√2*2)

This means that s√2=8√2 so side of square=8 and the area=64(8^2).

Since the diagonals of S intersect at the point (6, 6), we can say that (6,6) is the CENTER of the square.

Notice that, to get from point (2,2) to the CENTER at (6,6), we must travel up 4 units and then travel to the right 4 units.



Since (6,6) is the CENTER of the square, we can find another vertex by starting at the center and travelling 4 units up and then 4 units to the right



Now that we have our second vertex, we can see that the entire square looks like this:



Since each side of the square has length 8, the area of the square = (8)(8) = 64

We get:
QUANTITY A: 64
QUANTITY B: 64

Answer: C

Cheers,
Brent
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An alternative is to use the midpoint formula. The midpoint formula is the average of my x value and an average of my y values. I never teach my students the distance formula as it's always easier to create a right triangle. Instead, knowing that 6,6 is the midpoint of the diagonal I know that (x+2)/2 is 6, therefore X must be 10. Repeat for Y and you get 10,10 as the other vertice. Then you can draw that right triangle to get 8x8.
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You could also note that the diagonals intersect at the center of the square, so (6, 6) is the center. With (2,2) being one corner, the distance from the corner to the center is half of the square side so 6-2=4 is half the square side. The square sides are 2*4=8 and the area of the square is 8*8=64 as others have noted.