Here are two similar triangles. There are two possible values of
𝑥. Find both these values. State any assumptions that you make
when calculating each value.
For two shapes to be similar, they
must have the same angles. One will be an enlargement of the
other. When dealing with similar shapes
then, it’s useful to find the scale factor of the enlargement.
Here, however, there are two
possible values for the scale factor. We don’t know whether side 𝐴𝐵 in
the first shape corresponds to 𝐹𝐷 or 𝐸𝐷 in the second. So we’ll have to perform two
calculations depending on the assumption we make.
Let’s assume first then that 𝐴𝐵
corresponds to 𝐹𝐷 in the enlarged shape. The scale factor is found by
dividing the enlarged length by its corresponding original length. In this case, that’s 10 divided by
six. Since the length 𝑥 is on the
original shape, we will divide eight by the scale factor of enlargement to get back
to the original length. We can use the rules for dividing
fractions by changing the divide to a times and finding the reciprocal of 10 over
six, which is six tenths. Eight multiplied by six is 48. And one multiplied by 10 is 10. 48 divided by 10 is therefore
4.8. One value of 𝑥 is 4.8.
Next, we’ll assume that 𝐴𝐵
enlarges to 𝐸𝐷. In this case, the enlarged length
is eight and the corresponding original length is six. Its scale factor is
eight-sixths. Again, we divide the enlarged value
by the scale factor to get back to the original length 𝑥. 𝑥 is equal to 10 over one
multiplied by six over eight, which gives us 60 over eight.
We can simplify this fraction by
first dividing through by four, giving us 15 over two. 15 over two is 7.5. If the side 𝐴𝐵 enlarges to 𝐹𝐷,
𝑥 is equal to 4.8. If, however, 𝐴𝐵 enlarges to 𝐸𝐷,
𝑥 is equal to 7.5.