Step by step
The question
Question: D E is parallel to B C, with AE = 6, AC = 9, DE = 8: find BC.
1 Use the parallel line
D E is parallel to B C: the arrows show it.
So triangle A D E has the same angles as A B C: they are similar.
The small triangle is the big one made smaller: every side by the same scale.
2 Write the proportion
So: AE over AC equals DE over BC.
Careful!
Keep the same order on both sides: the matching parts on top, the matching parts at the bottom.
Put the numbers in.
We call BC x: 6/9 = 8 over x.
3 Cross multiply
When two fractions are equal, we can cross multiply.
These two are equal.
Work out the multiplications.
4 Remove the 6
6x means 6 times x.
We want just x.
To remove times 6, we divide by 6.
We do it on both sides.
Why both sides?
The = sign means both sides are the same amount.
If we change only one side, they are not the same any more.
6 divided by 6 is 1.
So just x is left.
72 divided by 6 is 12.
So x = 12!
5 Check
Let's check: put 12 in place of x.
First the left side.
Now the right side.
Both sides are 2/3.
So x = 12 is right!
So BC = 12.
Let's look back.
First: use the parallel line.
Then: write the proportion.
Then: cross multiply.
Then: remove the 6.
Last: check.