Finding Distance with the Pythagorean Theorem

Do you know that I love ice cream?

When I was young, there used to be an ice cream shop around the corner.

When I say around the corner, what I mean is down the street, around the corner and up the next street.

Kind of like this (above).

and I used to say to my friend ‘let’s go get some ice cream’ and he’d say that he’d race me for it.

So, I’d huff and puff and run all the way to the ice cream store.

When I’d get there, my friend would already be there, cone in hand.

It upset me so much.

It turned out that while I was running all the way around the corner and up to the ice cream, he was just cutting across the block.

We know that the shortest line between two points is a straight line between those points.

we know that he was going a shorter distance than I was.

But, how can we find out how much shorter of a distance?

well We use the Pythagorean Theorem .

The Pythagorean Theorem says that for a right triangle ( a and b are perpendicular to each other and c for the hypotenuse that a ^2 + b ^2 = c ^2.

so Let’s use the Pythagorean Theorem to find out how much further I had to go.

so here’s my block, the start and the ice cream.

now Each block has a length s .

So, I travelled s from start to the corner and s from the corner to the ice cream.

so These correspond to the distances a and b on a right triangle.

If I use the Pythagorean Theorem ( a ^2 + b ^2 = c ^2), I plug in this distance s (how far I had to go to the corner, and how far from the corner to the ice cream store) for a and b , I get s ^2 + s ^2 = c ^2.

now The c ^2 is the hypotenuse and is how far my buddy had to go in a straight line from the start to ice cream.

so I can solve this 2 s ^2 = c ^2.

and I can solve it for c ^2 - he travelled the square root of 2 s distance .

now Let’s compare that to what I travelled.

I travelled s to the corner plus s to the ice cream.

I went 2 s and he went the square root of 2 s .

This means that I travelled the length of two blocks, while he had to go less than a block and a half (about 1.4 blocks).

Let’s consider another case where we might want to use the Pythagorean Theorem.

I’m skateboarding down a ramp.

That ramp that I slide down is 20 ft long if i just slide down it, and it’s 6 ft tall.

So, how far out does the ramp go?

Again, let’s use our a ^2 + b ^2 = c ^2.

but now My a , in this case, is my height (6 ft).

The b I don’t know because I don’t know how far I’m travelling perpendicular to my height.

but I do know the hypotenuse and that’s 20 (I’m travelling down a 20 ft ramp).

So, let’s plug in 20 for c .

so I get 6^2 + b ^2 = 20^2.

I can solve this out, and I find that b is 19 ft.

so Let’s say you’ve got a different kind of triangle.

Here, you’ve got the Lion King peak.

You want to find out how high up the Lion King peak is and how much it overhangs.

so If I draw this out, I’m looking for this distance (the overhang distance), as well as the height.

Let’s say I know a little bit about this big rock though.

I know that it’s 4 ft across, 5 ft this way and about 8 ft this way.

How can I use the Pythagorean Theorem here?

I don’t actually have two right triangles, so how can I find the overhang and the height of this Lion King peak?

now I know that I have a right triangle made up of one length that I know (the height and the overhang), but in this case I have two unknowns and only one equation.

So, is there another triangle that I can use?

well Let’s write out our first equation.

Here’s our overhang that we’re going to call x , the height y and this hypotenuse for this triangle is actually 5.

So, I have x ^2 + y ^2 = 5^2.

Then, I’m going to draw a second triangle and I’m still going to use the overhang height, but now I’m going to include the entire Lion King peak in this example.

Now, my hypotenuse is 8 .

I know that one side is y , but what’s the other side?

What’s the length of this side here ?

what’s going to be 4 + x , because x is this length here (so that’s the overhang).

so that’s the width of my Lion King peak plus the amount that it overhangs.

well Let’s plug all that in: ( x + 4)^2 + y ^2 = 8^2.

well That’s great.

I have two equations and two unknowns.

This is just an algebra problem.

so I’m gonna solve this by subtracting my little triangle equation from my other equation.

That cancels out the y terms. I end up with an equation I can simplify as 8 x + 16 = 39 or x = about 2,8 2,9.

Okay, I’ve got x , now I just need to solve for y .

so I’m gonna use what I know for x for the overhang, which is x ^2 + y ^2 = 5^2.

Plug in x , square it, add y ^2 (which is unknown) equals 5^2.

I solve that, and I end up with y equaling just about 4,1.

so What I’ve found out is by using the Pythagorean Theorem twice on two different right triangles, my rock has a height of 4.1 ft and an overhang of about 2.9 ft.

alright, Let’s review.

the Pythagorean Theorem

We use it with right triangles.

so That’s where the two short lengths ( a and b ) are perpendicular to one another, and c is the hypotenuse.

The Pythagorean Theorem says a ^2 + b ^2 = c ^2.