Right angle at A, AD ⊥ BC, AB = 6, BC = 12, 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 AB = 6, BC = 12: 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 A B C: B D over A B equals A B over B C.

Cross-multiplied, that gives the rule: AB squared is BD times BC.

Careful!

It is BD times BC, not plus.

3 Put the numbers in

Put the numbers in.

We call BD x.

6 squared is 36.

4 Remove the 12

To remove times 12, we divide both sides by 12: 3.

5 Check

Let's check with Pythagoras in the small triangle A B D: A D squared plus B D squared is 36 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 12.

Last: check.