The vertices of quadrilateral ABCD are (A(1,1), B(1,5), C(5,5) and D(7,1). You want to transform ABCD into a parallelogram by only moving point B. A parallelogram is a four-sided figure with both pairs of opposite sides parallel. What should be the new x-coordinate of point B?
As it stands now, A and D are on the same horizontal line, and B and C are on the same horizontal line.
However, we want to move only point B. So, if we keep BC parallel to AD, then we want to have BC the same length as AD, which is 6.
Right now, the length of BC is 5-1 = 4. So, to increase the length to 6, staying on the line y=5, we move B to (-1,5).
Now we have two parallel sides, AD and BC of length 6. Note also that now AB and CD are parallel and the same length, so we have a parallelogram.
To transform quadrilateral ABCD into a parallelogram by only moving point B, we need to ensure that the opposite sides of the quadrilateral are parallel.
In this case, we have two pairs of opposite sides: AB and CD, and BC and AD.
To determine if the sides AB and CD are parallel, we can calculate their slopes. The slope of a line passing through two points (x₁, y₁) and (x₂, y₂) is given by the formula:
m = (y₂ - y₁) / (x₂ - x₁)
Let's calculate the slopes of AB and CD:
Slope of AB:
m₁ = (5 - 1) / (1 - 1) = 4 / 0 (undefined)
Slope of CD:
m₂ = (1 - 5) / (7 - 5) = -4 / 2 = -2
Since the slope of AB is undefined (division by zero), AB is a vertical line. For a parallelogram, the opposite sides need to be parallel, which means their slopes should be equal.
To make AB parallel to CD, we need to ensure that the new line segment AB has the same slope as CD, which is -2.
Given that the x-coordinate of point B needs to change while keeping the y-coordinate fixed, we can determine the new x-coordinate of B by finding a point on BC that has a slope of -2.
Slope of BC:
m₃ = (5 - 5) / (5 - 1) = 0 / 4 = 0
Since the slope of BC is 0, the line segment BC is a horizontal line. The slope of a horizontal line is always zero, and since we want AB and BC to be parallel, we need to find another point on the line BC such that its slope is -2.
Let's consider point E(x, 5) on BC, where x is the new x-coordinate of B.
Slope of BE:
m₄ = (5 - 1) / (x - 1) = 4 / (x - 1)
For AB and BC to be parallel, the slope of BE needs to be -2. Therefore, we can set up the following equation and solve for x:
4 / (x - 1) = -2
Multiply both sides of the equation by (x - 1):
4 = -2(x - 1)
Expand and simplify:
4 = -2x + 2
Subtract 2 from both sides:
2 = -2x
Divide both sides by -2:
x = -1
Therefore, the new x-coordinate of point B should be -1.