Head First 2D Geometry

mind the gap

What’s the longest the rope can be without going below the

safety line?

4m + 2m = 6m

Safety line

The floor

4m

2m

2m

T

his 2m is below the safety line, so

not included

Onl

y add up the

heights ABOVE the

safety line.

So, how far can you swing on a six-meter rope?

The gap between the platforms is the base of a triangle, with the rope

making up two of the sides joined at the point where the rope is fixed at the

top. So the distance of the gap is the same as the length of the base of the

triangle.

W

e need t

o find the

dis

tance of this gap.

T

his angle isn’t

a r

ight angle.

Safety line

The floor

4m

2m

d

142 Chapter 3

the pythagorean theorem

Well, the rope doesn’t change

length, does it? So I guess that gives

us two sides of the triangle, but we

can’t use the Pythagorean Theorem

to find the missing side without a

right angle, can we?

How can you make the swing problem into a

right triangle problem so that you can use the

Pythagorean Theorem to find length d?

Safety line

The floor

4m

2m

d

6m

That’s right—the Pythagorean Theorem

only finds missing sides for right

triangles.

So, when you’re faced with a triangle without a right

angle you’ve got two options: find something else in

your Geometry Toolbox to solve the problem, or see if

you can somehow turn your non-right triangle into a

right triangle (or triangles!)

you are here 4 143

BISECTS this line.

Bisec

t

create your own right triangles

How can you make the swing problem into a right triangle problem

so that you can use the Pythagorean Theorem to find d?

Solution

The base of the triangle is horizonal, so

a vertical line down from the top of the

triangle splits it into two equal right

triangles.

Safety line

The floor

4m

2m

d

6m

You can split an isoceles triangle into two

congruent right triangles

You can split a triangle into two right triangles by drawing an

altitude—a line which joins the top of the triangle to the base,

perpendicular to the base. An isoceles triangle has two sides and two

angles the same, so the two triangles created are congruent.

6m 6m6m 6m

W

Two sides the same

Two angles the same

T

his line is called

an a

lti

tude.

T

he alti

tude

BISECTS

this angle.

T

he alti

tude

Chop int

o t

wo

equal parts.

W

144 Chapter 3

the pythagorean theorem

BE the Rope Swing

Your job is to play like you’re the 6m

rope. Use the Pythagorean Theorem to

work out how far across a gap someone

could swing on you.

Q:

I get that the rope length doesn’t

change, so the sides are equal, but how

did we know that the bottom two angles

are equal?

A:An isoceles triangle has two sides

equal but also two angles equal—always.

So—if you see that the sides are the same

you know the angles are the same, and the

other way around.

Q:

How did we know that the altitude

Q:

So, which triangle does the

would be vertical? altitude belong to?

A:The altitude is always perpendicular A:Both! The altitude is the shared side

to the base that it’s drawn on, so if that of the two identical (congruent) triangles it

base is horizontal (as it is in this case), creates in this case. So, it belongs to both

then the altitude must be vertical. of them.

you are here 4 145