Following
IanStewart Using Coordinate Geometry to solve this question is doable so long as you understand the reasoning behind Reflected Points and the Mirror Line.
Setting up the Square in the Coordinate Plane as Ian said to do:
A - (0 , 10)
E - (0 , 5)
D - (0 , 0)
C - (10 , 0)
B - (10 , 10)
Let Point F be ----> (X , Y)
if we can find the Y Coordinate for Point F, we will know the Perpendicular Distance to the X-Axis, which the Side CD lies upon.
By folding over the Vertex A -----> to Point F, we are effectively taking Point A (0 , 10) and reflecting this Point over the Line that Passes through Point E and Point B
(1st) Find the Equation of the Mirror Line - the line over which we are Reflecting Point A
If set up correctly in the Coord. Plane, Line EB will pass through Point E (0 , 5) and Point B (10 , 10)
m = Slope = +1/2
b = Y Intercept = 5
Formula for Mirror Line: y = (1/2)x + 5
(2nd) There are 2 Concepts that can help make the calculation easier:
-1- If we draw in a Line Segment connecting Original Point A and its Reflected Image Point F, the Line given by y = (1/2)x + 5 will BISECT this segment.
In fact, every Point on the Mirror Line will be Equidistant from Points A and F.
-2- the Slope of our created Line Segment AF will have a Slope that is the Negative Reciprocal of +(1/2). This is because the Mirror Line will be Perpendicular to our created Line Segment AF. (this works out because Point A is on the Y Axis at (0 , 10))
(3rd) We can use the above 2 Concepts and create 2 Equations to solve for Point F (X , Y)
-1- Find the Mid-Point of Line Segment AF and then Plug this Mid-Point into the Mirror Line's Equation of: y = (1/2)x + 5
Mid-Point is:
X Coordinate = (0 + X)/2 = X/2
and
Y Coordinate = (10 + Y)/2
Plugging the Mid-Point's Coordinates into the Mirror Line's Formula: y = (1/2)x + 5
(10 + Y)/2 = (1/2)(X/2) + 5
Solving this Equation ------> X = 2Y
-2- (Slope of Mirror Line) * (Slope of Line AF) = -1
Slope of Mirror Line = +(1/2)
Slope of Line AF (again Letting Point F have the Coordinates (X , Y) =
(10 - Y) / (0 - X) = (10 - Y) / (-X)
Plugging the Slopes into the Equation above:
[ (10 - Y) / (-X) ] * [ (1/2)] = -1
----multiplying both sides by +2-----
(10 - Y) / (-X) = -2
10 - Y = 2X
From -1- we found that: X = 2Y -------> Substituting in for X
10 - Y = 2*(2Y)
10 - Y = 4Y
10 = 5Y
Y = 2 = the Y Coordinate of Point F in the Coordinate Plane
Since X = 2Y, we can also find the X Coordinate of Point F ------> X = 4
Thus, Point F will be at (4 , 2)
and this Point will be exactly 2 Units above the X-Axis, which shares Side CD in our Graphed Square.
Answer = 2
-B-