Step by step
The question
Question: A D cuts angle A in half, with AB = 6, BD = 3, DC = 5: find AC.
1 Use the bisector rule
A D cuts the angle at A into two equal angles.
It is the angle bisector.
It cuts B C in the same ratio as the two sides next to A: B D to D C is like A B to A C.
2 Write the proportion
So: AB over AC equals BD over DC.
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 AC x: 6/x = 3/5.
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 3
3x means 3 times x.
We want just x.
To remove times 3, we divide by 3.
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.
3 divided by 3 is 1.
So just x is left.
30 divided by 3 is 10.
So x = 10!
5 Check
Let's check: put 10 in place of x.
First the left side.
Now the right side.
Both sides are 3/5.
So x = 10 is right!
So AC = 10.
Let's look back.
First: use the bisector rule.
Then: write the proportion.
Then: cross multiply.
Then: remove the 3.
Last: check.