Right angle at A, AD ⊥ BC, AD = 12, DC = 16, find BD

Step by step

The question

Question: triangle A B C has a right angle at A, and A D is at 90° to B C, with AD = 12, DC = 16: find BD.

1 Find the similar triangles

The height A D cuts the big triangle into two small ones.

All three triangles have a right angle, and they share their other angles.

So they are similar: the same shape.

2 Write the rule

Triangles D B A and D A C: B D over A D equals A D over D C.

Cross-multiplied, that gives the rule: AD squared is BD times DC.

Careful!

It is BD times DC, not plus.

3 Put the numbers in

Put the numbers in.

We call BD x.

12 squared is 144.

4 Remove the 16

To remove times 16, we divide both sides by 16: 9.

5 Check

Let's check with Pythagoras in the small triangle A B D: A D squared plus B D squared is 225 the same as A B squared.

So it fits!

Let's look back.

First: find the similar triangles.

Then: write the rule.

Then: put the numbers in.

Then: remove the 16.

Last: check.