Question

On the accompanying grid, draw and label quadrilateral ABCD with points A(1,2), B(6,1), C(7,6), and D(3,7). On the same set of axes, plot and label quadrilateral A′B′C ′D′, the reflection of quadrilateral ABCD in the y-axis. Determine the area, in square units, of quadrilateral A′B′C′D′.

Answer 1

Answer: Refer to figure 1 for the diagram needed.

A ' = (-1, 2)

B ' = (-6, 1)

C ' = (-7, 6)

D ' = (-3, 7)

Area = 24 square units

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Explanation:

Start by drawing an xy axis. Plot the given points A,B,C,D on this grid.

To reflect any given point over the y axis, we apply this rule

$(x,y) \to (-x,y)$

which means we change the sign of the x coordinate, but keep the y coordinate the same. For example, a point like A(1,2) moves to A ' (-1,2). The other points are handled in a similar fashion.

After all of the points have been reflected, you'll get what is shown in figure 1.

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To find the area of quadrilateral ABCD, we can form a bounding box as small as possible around the quadrilateral. Refer to figure 2.

The area of rectangle FEHG is 6*6 = 36 square units

Then we subtract off the areas of these four outer triangles

• Triangle AED = 5*2/2 = 5
• Triangle AFB = 1*5/2 = 2.5
• Triangle BCG = 1*5/2 = 2.5
• Triangle CDH = 1*4/2 = 2

Meaning we'll end up with 36-5-2.5-2.5-2 = 24 as the area of quadrilateral ABCD. Because a reflection is a mirror copy, quadrilateral A'B'C'D' will have the same area as ABCD.