I have a rectangle whose sides are in the proportion b:a, where b is the longer side. I draw a line parallel to a inside the rectangle in order to break it up into a square with sides of length a and a second rectangle. Amazingly, the sides of the new smaller rectangle are in the exact same proportion as my original rectangle, except that a is the longer side.
Draw a picture and solve for the proportion b:a. (Note that if you set a equal to 1, b automatically equals the proportion) You will need to use the quadratic formula. The formula will give you two answere, but only the positive answer is relevant.
To solve this problem, let's start by drawing a rectangle with sides b and a.
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Next, draw a line parallel to the side with length a. This line will divide the rectangle into a square with sides of length a and another rectangle.
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Let's label the width of the smaller rectangle as x. Therefore, the longer side of the smaller rectangle is a + x.
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Now, we need to establish the proportion between the sides of the original rectangle and the smaller rectangle. The original rectangle has sides in the ratio b:a, and the smaller rectangle has sides in the ratio x:(a + x).
Using this information, we can set up the following proportion:
(b/a) = (x/(a + x))
To solve for x, we can cross-multiply:
b(a + x) = ax
Expand the equation:
ab + bx = ax
Rearrange the equation:
ax - bx = ab
Factor out x:
x(a - b) = ab
Now, solve for x:
x = ab / (a - b)
Since a is the longer side in the smaller rectangle, a > b. Thus, a - b > 0, which means we can use the quadratic formula to solve for x.
The quadratic formula is given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 1, b = -(a - b), and c = -(ab).
Substituting these values, we have:
x = (-(a - b) ± √((a - b)^2 - 4(1)(-ab))) / (2(1))
Simplifying further:
x = (-(a - b) ± √((a - b)^2 + 4ab)) / 2
Calculating the discriminant (b^2 - 4ac):
(a - b)^2 + 4ab = a^2 - 2ab + b^2 + 4ab = a^2 + 2ab + b^2
Substituting the discriminant into the equation:
x = (-(a - b) ± √(a^2 + 2ab + b^2)) / 2
x = (-(a - b) ± √((a + b)^2)) / 2
x = (-(a - b) ± (a + b)) / 2
Simplifying further:
x = (b - a ± (a + b)) / 2
There are two possible solutions:
1. x = b / 2 (positive answer)
2. x = -2a (negative answer, not relevant since width cannot be negative)
Therefore, the proportion b:a is given by b/2:a.