rectangle split
Draw another line through the centre of this rectangle to split it into 4 pieces of equal area.
Problem
The line drawn through the centre of this rectangle splits it into two pieces of equal area.
Another line is drawn through the centre to split the rectangle into four pieces of equal area. What lengths is the 10 cm side split into?
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Student Solutions
Using triangles
In the diagram below, horizontal and vertical dotted lines are also drawn through the centre of the rectangle. The two red triangles are congruent and so are the two yellow triangles, and the length $x$ needs to be chosen so that the area of the red triangles is equal to the area of the yellow triangles.
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That means that $5\times1$ must be the same as $6\times x$, so $5=6x$, so $x=\dfrac{5}{6}$.
So the $10$ cm side is split into lengths of $5+\dfrac{5}{6}=5\dfrac{5}{6}$ and $5-\dfrac{5}{6}=4\dfrac{1}{6}$.
Using scale factors
Imagine if instead we had a square, as shown below. Then the sides should all be split in the same ratio to give four equal (and congruent) areas.
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Stretching the square will give us a rectangle, and a stretch will preserve the ratio between the areas.
So starting with a $12$ cm by $12$ cm square, a vertical stretch with scale factor $\dfrac{10}{12}=\dfrac{5}{6}$ will give us the desired rectangle, and the $10$ cm side will be split into pieces of length $\dfrac{5}{6}a$ and $\dfrac{5}{6}b$ cm.
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In this case, $a$ and $b$ are $7$ and $5$, so $\dfrac{5}{6}a=\dfrac{5\times7}{6}=5\dfrac{5}{6}$ and $\dfrac{5}{6}b=\dfrac{5\times5}{6}=4\dfrac{1}{6}$ cm.