Bunuel wrote:

In the figure above, ABCE is a square. What are the coordinates of point B?

(A) (-4,2)

(B) (-2,4)

(C) (-2,6)

(D) (4, -6)

(E) (6,-2)

We can get the answer without calculating anything except two obvious lengths,

using either elimination or the "box" method

Eliminate:

• Answers D) (4, -6) and E) (6,-2). Coordinates of B are in Q II (-x,y). D and E have (x, -y)

• Answer A: (-4, 2) is the mirror point for C (4, 2). Not allowed. Square is not symmetric about the axes.

If you were to flip C across the y-axis, its mirror point would be (-4, 2)

C's mirror point across the y-axis cannot be a vertex because the square is not symmetric about the axes.

• Answer B: (-2,4). Distance of B from x-axis MUST be greater than distance of D from x-axis. More than half of the square is in QI and QII.

The y-coordinate of B (the height of B from the x-axis) MUST be greater than D's distance from x-axis. D's distance is 4. B must be > 4

That leaves one answer: (-2, 6)

Answer CAttachment:

square2018.07.25.jpg [ 35.18 KiB | Viewed 1008 times ]
The "box method"If geometric figures are not parallel to the x- and y-axes,

draw a box around the figure.

Now we have congruent right triangles all around the smaller square.

The side lengths of those triangles are easy to find because they are right triangles whose vertical and horizontal legs can be measured with the x- and y-axes.

From

line segment AO = 6 and properties of a square (parallel sides) we know that one segment of a side of large blue-edged square = 6

From

line segment DO = 4 and properties of a square we know that the other segment of a blue-edged side = 4

A square inscribed in a square divides the sides of the larger square proportionally (Pythagorean theorem)

The larger square's sides are in segments of length 6, 4

Right triangle ABX has height 6. By properties of a right triangle (perpendicular legs), point X and point B are collinear.

The

y-coordinate of B is

6. That leaves one answer.

(-2, 6)Answer CIf you wanted to ascertain the x-coordinate, keep in mind the larger square side's a, b, a, b pattern

By properties of a square, Points C and Y MUST have the same x-coordinate.

Both are 4 away from the y-axis.

But segment BY must have length 6 (by property of inscribed square)

In QII, then, the distance of B from the y-axis is (6-4) = 2

Because the vertex is in QII, its x-coordinate is (

-2, 6)

_________________

In the depths of winter, I finally learned

that within me there lay an invincible summer.

-- Albert Camus, "Return to Tipasa"