How to Solve for Invisible Shapes

We’ve discussed how drawing pictures can help you solve math problems. But what happens when we can’t see what you are looking for?

Shorten the trip

If Diane were able to travel from A to D directly, how much shorter would her trip be?

 

 

 

 

 

 

 

 

As always, draw a picture so you can visualize the problem.

 

 

 

 

 

 

 

 

Dr. Hypotenuse, I presume?

My students clearly see that they need to solve for the hypotenuse.

 

But we need to know at least two sides in order to then solve for the hypotenuse.

We don’t have enough information to figure out the length of the two pieces of CB.

 

 

 

 

 

The invisible triangle strikes again!

The orange line is a hypotenuse – of an invisible triangle.

 

 

 

 

 

 

 

 

 

 

Now we need to figure out the lengths of the sides.

 

 

 

 

 

 

 

Aha! We can find the hypotenuse of this triangle using a^2 + b^2 = c^2.

(Bonus points if you recognize that it is a variation of a 3-4-5 triangle. Therefore the hypotenuse must be 25.)

 

 

 

 

 

 

Now that we’ve figured out the length must be 25, go back and check your map. It is 35 miles taking the prescribed route, so 35-25 = 10. Taking the shortcut is 10 miles shorter.

Image source: vaXzine on Flickr