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Chapter 9. triangles, trig and trajectories

Chapter 9. triangles, trig and trajectories

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castle defense needed



Camelot - we have a problem!

The Head First castle is in imminent danger! Back when it was

built, the longest ladder available from Sieges-R-Us was 15.0 m

long. So the castle was designed with a moat 15.0 m wide and

a wall 15.0 m high, making it impossible for anyone to put a

ladder from the edge of the moat to the top of the wall.



End of ladder

is nowhere near

top of wall.



Ladder



15.0 m



Moat filled

with water



Bottom of

ladder is at

edge of moat.



15.0 m



Wall



15.0 m



But the Sieges-R-Us website has just been

updated with a new top of the range 25.0 m

ladder. It’s only a matter of time before someone

comes to attack your castle armed with the new

ladder, and your current defense system just isn’t

big enough...



New ladder will

comfortably reach

top of wall!



Wall



25.0 m



Ladder



Moat



15.0 m



15.0 m



The new ladder



If you don’t act quickly, someone will turn up

with the new ladder, and you’ll be toast. It’s time

to design a new castle defense system!



336   Chapter 9

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triangles, trig, and trajectories



Check the box

next to the idea

you think is best.



Here are some ideas for a new castle defense system.

Unfortunately, you don’t have many spare stones lying around, only

some shovels and food.

Write down at least one advantage and disadvantage of each idea,

and check the box next to the idea you think is the best.

Coat the top of the wall with something slippery.

Advantage(s)



Disadvantage(s)



Make the moat wider.

Advantage(s)



Disadvantage(s)



Run away! Run away!

Advantage(s)



Disadvantage(s)



Make the wall higher.

Advantage(s)



Disadvantage(s)



Put a health and safety rep on the top of the wall to recite the

working height directive repeatedly to anyone who gets higher

than 2 m. Also to ask when their ladders were last inspected

and if they have been trained in the proper use of ladders.

And insist that they all wear safety harnesses, hard hats, ear

defenders, goggles, gloves and toe protectors.

Advantage(s)



Disadvantage(s)



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we need a wider moat



Here are some ideas for a new castle defense system.

Unfortunately, you don’t have many spare stones lying around, only

shovels and food.

Write down at least one advantage and disadvantage of each idea,

and check the box next to the idea you think is the best.

Coat the top of the wall with something slippery.

Advantage(s)



Ladders will slip, it might

delay them a bit.



Disadvantage(s)



Washes off in the rain, attracts

cats, not a long-term solution.



Make the moat wider.

Disadvantage(s)



Advantage(s)



I’m not sure how wide to make

the moat.



I have a shovel to dig a

wider moat.



Run away! Run away!

Advantage(s)



I won’t be there when they

get into the castle.



Disadvantage(s)



They’ll get into the castle.



Make the wall higher.

Advantage(s)



This would definitely keep

them out of the castle ...



Disadvantage(s)



... but there’s no stone available

to build a higher wall.



Put a health and safety rep on the top of the wall to recite the

working height directive repeatedly to anyone who gets higher

than 2 m. Also to ask when their ladders were last inspected

and if they have been trained in the proper use of ladders. And

to Insist that they all wear safety harnesses, hard hats, ear

defenders, goggles, gloves and toe protectors.

Advantage(s)



The attackers might get bored

and attack a different castle.



Disadvantage(s)



By wearing ear defenders, they

can ignore him and continue

attacking the castle anyway.



338   Chapter 9

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It’s a lot easier to

dig a wider moat

than it is to build

a higher wall.



triangles, trig, and trajectories



How wide should you make the moat?

Widening the

moat moves the

bottom of the

ladder further

away from the

top of the wall.



The best way of defending the castle against the

25.0 m ladder is to make the moat wider. The

moat is already 15.0 m wide—so how much

wider do you need to make it?

You could try making the moat the same size as

the ladder —25.0 m— so that the distance from

the edge of the moat to the bottom of the wall is

the same length as the ladder. That would make

sure that attackers couldn’t simultaneously put

one end of the ladder at the edge of the moat

and the other end on the top of the wall.



You want the distance from

the edge of the moat to

the top of the wall to be

more than 25.0 m.



15.0 m



? m



Old moat width

But time is of the essence, and you don’t want to start out

digging a 25.0 m moat if a narrower moat will do the same

job. The important thing is the distance from the edge

of the moat to the top of the wall. If that’s more than

25.0 m, there won’t be anything to lean the ladder on. And

you might be able to achieve that with a narrower moat... a

sketch should help.



a. Draw a sketch of the 15.0 m castle

wall, 25.0 m ladder, and extended

moat, where the ladder is only just

too short to reach the top of the

wall from the side of the moat.

(This sketch is just a quick drawing

to get the visual parts of your brain

working—the lengths on it don’t

have to be accurate as long as

everything’s labelled correctly.)



b. What shape does your sketch resemble?



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remember triangles?



Top of ladder doesn’t

quite reach top of wall.



a. Draw a sketch of the 15.0 m castle

wall, 25.0 m ladder, and extended

moat, where the ladder is only just

too short to reach the top of the wall

from the side of the moat.



Ladder

0 m



Minimum width of moat = ? m

This distance is less than

25.0 m. So this way is quicker

and more efficient than

making the moat 25.0 m wide.



b. What shape does your sketch resemble?



It looks like a triangle.



You can turn your complicated-looking castle, ladder,

and moat sketch into a more simple picture of a

right‑angled triangle, with a 90° angle (a right

angle) between the wall and the moat. You already

know the lengths of two of the triangle’s sides and

want to find out how long the third side is.



Simplified version

of our sketch



Ladder

25.0 m

15.0 m



You could figure this out by ordering a 25.0 m

ladder, putting one end at the top of the wall, and

seeing where the ladder touches the ground. But the

attackers might arrive with their new ladder first!

If you don’t want to wait, you can measure 25.0 m

of rope, tie one end to the top of the wall, and see

where it touches the ground when you pull it tight.

But that still involves a lot of steps and equipment.



15.0 m

Moat



Bottom of ladder is

right at edge of moat.



Looks like a triangle, yeah?



Wall



25.



(This sketch is just a quick drawing

to get the visual parts of your brain

working—the lengths on it don’t have

to be accurate as long as everything’s

labelled correctly.)



Wall



Moat

? m



You need to

work out how

long this side is.



340   Chapter 9

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Dawr a little square inat

the corner to show th E.

this is a RIGHT ANGL



triangles, trig, and trajectories



Wouldn't it be dreamy if you

could just work out the smallest moat

width from the triangle drawing, without having

to climb the wall and make a lot of measurements?

But I know it's just a fantasy…



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scale drawings



A scale drawing can solve problems

This is one way of getting

the best moat width from

a triangle drawing. But it’s

not the best way.



At the moment, your drawing is only a sketch—the

triangle’s side lengths aren’t to scale (though the

lengths you’ve written beside them are correct).

A scale drawing is one where you say something like

“1 cm on the drawing = 1 m in real life.” You can then

do the same as you would with the 15.0 m castle wall

and 25.0 m ladder, except with 15 cm and 25 cm!



Start off by making a right-angle to represent where the wall

meets the moat. Then measure up 15 cm to represent the wall.

Now, you want to swing the ladder down and see where it hits

the ground. So set a pair of compasses to 25 cm...



A scale drawing takes

time, and I’m sure someone

must have solved a problem

like this before. Is there an

equation that’ll help?



m

25 cwn).

o

t

s

o

passeladder d

m

o

c

of

he

pair winging t

a

t

Se e as s

(sam

Make a right-angle here.



Measure up

15 cm for

the wall.



... and swing down from the top of the wall to meet the ground

line. Now you can measure the most economical moat width.

That’s a lot of effort just to work out a simple length though ...



Sweep compasses

around until you cross

the horizontal line.



You can solve some problems

with scale drawings, but it

takes time and effort.

342   Chapter 9



25



cm



15 cm



? cm

Accurate scale version

of your triangle.



Measure this side and

scale it up to get the

best moat width.



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triangles, trig, and trajectories



Pythagoras’ Theorem lets you figure out the sides quickly

Pythagoras’ theorem is an equation for solving this

kind of problem without waving ladders around or

making a super-accurate drawing. You only need to

know the lengths of two sides, and the equation will tell

you the third.



Longest side

is called the

hypotenuse.



The longest side of the triangle is opposite the biggest

angle (the right angle). This side has a special name and

gets called the hypotenuse.



The hypotenuse is

opposite the right angle

- the largest angle.



If you square the length of the hypotenuse, the answer is

equal to the answer you get if you square the length of the

other two sides individually, then add the squares together.



c



That’s very wordy. So here’s the equation - if you label the

sides of your triangle a, b and c (where c is the hypotenuse)

then Pythagoras’ Theorem says:





c2 = a2 + b2



b



It doesn’t really matter which

letters you use for the sides.

We’ve chosen the same letters as

the AP physics equation table.



a



c2 = a2 + b2

The hypotenuse is on one side of

the equation, the rest are on the

other side of the equation.



Pythagoras’ Theorem only works for right‑angled

triangles. You can’t use it if your triangle doesn’t have a

right angle.



If you already

know two sides of

the right‑angled

triangle, Pythagoras

gets you the third.



Pythagoras only works for

right‑angled triangles.



If the triangle is right‑angled:

The square of the hypotenuse





is equal to the sum



of the other two sides squared:





c2 = a2 + b2

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look for right angles



Q:



You’ve called Pythagoras a theorem and an equation

so far. And I’ve seen things like that called a formula too. So

which is it?



A:



Equation, formula, and theorem mean the same thing really.

They all describe relationships where you write down “something =

another thing.”



Q:



Q:



What if I know the hypotenuse and want to calculate the

length of one of the other sides?



A:

Q:



You can rearrange the equation so that the side you don’t

know is on its own on the left.



So where does Pythagoras’ Theorem come from? Aren’t

we going to go through proving it?



How do I try to remember Pythagoras? I mean, how do

I remember which sides are a, b and c, then what order to put

them in the equation?



A:



If you can remember the form of the equation, you don’t

need to remember the letters. The hypotenuse is the longest side,

whatever letter you use to name it. So hypotenuse goes on the left

of the equation.



Being able to prove Pythagoras’ Theorem doesn’t help with this, so

we’ve not gone into that here.



A:



It’s only really worth going into understanding where an

equation comes from if the understanding you gain helps you see

how the world works, so you can solve physics problems (and other

problems) better.



And on the right of the equation, you square each of the other sides,

then add them together. You can think about the S in SUCK - size

matters. So it’s the square of the longest side that goes on its own.



We’ll probably see a lot of

right‑angled triangles in physics

because the ground’s horizontal and

gravity acts vertically?



You’ll spot many right angles between the

horizontal and vertical

Right‑angled triangles are going to be one of your most

important physics tools. There are lots of right angles in

physics, often between the horizontal ground and vertical

walls or vertical acceleration vectors that exist as a result

of gravity.

Keep your eyes open for them as this chapter progresses ...



344   Chapter 9

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triangles, trig, and trajectories



Sketch + shape + equation = Problem solved!

Back to the castle and the new Sieges-R-Us ladder!

You started with a sketch and spotted a right‑angled triangle shape in it. After toying

with the idea of a scale drawing, Pythagoras popped up with an equation!

So now you can work out the best moat width—and save the castle!



c2 = a 2 + b 2



Ladder

Wall



0 m



25.



15.0 m



25.0 m

15.0 m



c

b



Moat

Minimum width of moat = ? m



? m



Start with a sketch



Look for familiar shapes

(triangles, rectangles, etc)



a

Use an equation that tells you

about this kind of shape



Solve your problem!



A castle is built on flat ground with 15.00 m walls. How wide must the moat be to

ensure that a 25.00 m ladder only just touches the top of the wall? Assume that the

base of the ladder is placed at the edge of the moat.



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pythagorean theorem



A castle is built on flat ground with 15.00 m walls. How

wide must the moat be to ensure that a 25.00 m ladder

only just touches the top of the wall? Assume that the

base of the ladder is placed at the edge of the moat.



Start with a sketch:



Ladder



 m

5.0



Wall

15.0 m



2



or 32 = 3×3 = 9.



If you take the square root (√ ) of a number, the

answer you get is the number you’d have to

square to get the one you started off with. For

example, √9 = 3 because 32 = 9.



Want to know a, the width of the moat.

By Pythagoras, c = a + b



Rearrange

equation

for the side

you want.



A:





?m



2



There were square roots in that solution,

but it’s been a while since I used these, and

they’re a bit hazy. Remind me how they work

again?

As we saw in chapter 3, the square of a

number is the number times itself.



Moat



This is what you say

if you use Pythagoras.



Q:



2



2



a2 = c2 - b2



Q:



I noticed that I got nice round numbers –

15.0, 20.0 and 25.0 – for the side lengths. Does

that always happen with right‑angled

triangles?



A:



a2 = 252 - 152

a = 400

2



a = 400 = 20.0 m

So the best moat width is 20.0 m (3 sd).



Values in your question

were given to 3 sd, so

your answer should

have 3 sd too.



346   Chapter 9

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Here, the wall, ladder and moat formed a

right‑angled triangle which has nice side lengths

in a 3:4:5 ratio. 32 + 42 = 52, and a 15:20:25 ratio

is just a 3:4:5 ratio multiplied by 5.



But usually that doesn’t happen - your calculator

will give you answers that you’ll have to round

to the same number of significant digits as the

values you were initially given to work with.



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