Step by step
The question
Question: A D cuts angle A in half, with AB = 10, AC = 15, DC = 9: find BD.
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 BD x: 10/15 = x/9.
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 15
15x means 15 times x.
We want just x.
To remove times 15, we divide by 15.
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.
15 divided by 15 is 1.
So just x is left.
90 divided by 15 is 6.
So x = 6!
5 Check
Let's check: put 6 in place of x.
First the left side.
Now the right side.
Both sides are 2/3.
So x = 6 is right!
So BD = 6.
Let's look back.
First: use the bisector rule.
Then: write the proportion.
Then: cross multiply.
Then: remove the 15.
Last: check.