Step by step
The question
Question: D E is parallel to B C, with AD = 5, DB = 10, EC = 8: find AE.
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.
A parallel line cuts the two sides in the same ratio: A D to D B is the same as A E to E C.
2 Write the proportion
So: AD over DB equals AE over EC.
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 AE x: 5/10 = x/8.
3 Cross multiply
When two fractions are equal, we can cross multiply.
These two are equal.
Work out the multiplications.
Let's swap the two sides, so x is on the left.
4 Remove the 10
10x means 10 times x.
We want just x.
To remove times 10, we divide by 10.
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.
10 divided by 10 is 1.
So just x is left.
40 divided by 10 is 4.
So x = 4!
5 Check
Let's check: put 4 in place of x.
First the left side.
Now the right side.
Both sides are 1/2.
So x = 4 is right!
So AE = 4.
Let's look back.
First: use the parallel line.
Then: write the proportion.
Then: cross multiply.
Then: remove the 10.
Last: check.